Physica E 75 (2016) 28–32

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Physica E
journal homepage: www.elsevier.com/locate/physe

Directional dependence of vortex core resonance in a square-shaped
ferromagnetic dot
Xiaomin Cui a, Satoshi Yakata b, Takashi Kimura c,d,naGraduate School of Information Science and Electrical Engineering, Kyushu University, 744 Motooka, Fukuoka 819-0395, Japan
Department of Information Electronics, Fukuoka Institute of Technology, 3-30-1 Wajiro-higashi, Higashi-ku, Fukuoka 811-0295, Japanc
Department of Physics, Kyushu University, 6-10-1 Hakozaki, Fukuoka 812-8581, Japand
Research Center for Quantum Nano-Spin Sciences, Kyushu University, 6-10-1 Hakozaki, Fukuoka 812-8581, Japanb

H I G H L I G H T S

 The dynamic property of a magnetic vortex conﬁned in a square ferromagnetic dot has been investigated.
 The ﬁeld dependence of the resonant property strongly depends on the direction of the magnetic ﬁeld.
 The resonant frequency is strongly modiﬁed by the magnetic ﬁeld along the diagonal direction of the square.

art ic l e i nf o

a b s t r a c t

Article history:
Received 26 August 2015
Accepted 29 August 2015
Available online 3 September 2015

The resonant property of the magnetic vortex conﬁned in a square-shaped ferromagnetic dot has been
investigated. We showed that the ﬁeld dependence of the resonant frequency has a unique directional
dependence originating from a four-fold rotational symmetry of the square. The resonant frequency isfound to be strongly modulated by the magnetic ﬁeld along the diagonal direction although the magnetic
ﬁeld applied along the side of the square hardly modiﬁed the resonant frequency. The modulation ratioof the resonant frequency deﬁned by the ratio between minimum and maximum frequencies for thevortex resonance was found to be tuned by the lateral dimension of the square. These unique frequency
tunabilities controlled by the magnitude and the direction of the magnetic ﬁeld may provide additional
functions in the application of the magnetic vortex systems.
& 2015 Elsevier B.V. All rights reserved.

Keywords:
Vortex core resonanceSpin dynamics
Patterned magnetic structures
Microstrip line

1. Introduction
A magnetic vortex is a unique spin conﬁguration in a micron or
submicron-sized ferromagnetic dot and shows numerous intriguing and attractive properties both from the fundamental andtechnological view points. Robust thermal stability and low frequency dispersion are the representative advantages of the magnetic vortex [1–5]. Therefore, the magnetic vortex structures are
considered as the promising structures of the unit cell for the spinbased electronic devices such as magnetic random access memoryand spin–torque oscillators [2–5]. In addition to these attractive
properties, it is another important advantage that the statical anddynamical characteristics of the magnetic vortex can be adjusted
by the dot dimension and/or the conﬁguration of the dot array.
nCorresponding author at: Department of Physics, Kyushu University, 6-10-1
Hakozaki, Fukuoka 812-8581, Japan.
E-mail address: t-kimu@phys.kyushu-u.ac.jp (T. Kimura).

http://dx.doi.org/10.1016/j.physe.2015.08.038
1386-9477/& 2015 Elsevier B.V. All rights reserved.

Moreover, the biological applications such as cancer detector andvirus sensors have been demonstrated because of the nonresidual
magnetic moment at the remanent state with high magnetic
susceptibility [1,4,6]. Thus, the varieties of the application possibilities are expected by using the magnetic vortex systems.
The dynamic property of the magnetic vortex core can be described by Thiele's equation, where the trajectory of the vortex
core exhibits a rotational motion around its equilibrium position
about sub-gigahertz range [7–11]. The gyration motion of vortex
core offers abundant fundamental behaviors because the motion
equation of the core is mathematically equivalent situation for themass point conﬁned in the parabolic potential. However, Thiele's
equation is valid only in the motion without changing the structures of the magnetic vortex such as the conﬁned potential and thecore size. Therefore, the experimental characterization of the coremotion including the non-linear regime may deepen the understanding towards the basic property of vortex core and pave theway for improving the performance of the spin-based nanoelectronic devices using a magnetic vortex [10–13].

X. Cui et al. / Physica E 75 (2016) 28–32

Although most of the studies on the magnetic vortex have been
carried out by employing simple circular ferromagnetic disk [14–17], several unique features have been reported in the magnetic
vortex conﬁned in other shapes. For example, in a ferromagnetic
dot with a regular triangle shape, the chirality of the vortex corecan be well controlled and its resonant frequency is strongly
modiﬁed by adjusting the core position [18]. In a elliptical ferromagnetic disk, an anisotropic ﬁeld dependence of the core resonant frequency has been reported [19,20]. Thus, the resonant
properties of the magnetic vortex can be ﬂexibly modiﬁed bytuning the geometrical shape as well as their dimension [21,22].
These facts imply that excellent novel properties are anticipated by
exploring the magnetic vortex conﬁned in a speciﬁc-shaped ferromagnetic dot. For seeking new functionalities of magnetic vortex, in the present study, we investigate the core dynamics of themagnetic vortex conﬁned in a square-shaped ferromagnetic dot.

2. Method
We fabricated a chain of the square ferromagnetic dots on a
non-doped Si substrate by the conventional lift-off technique.
Here, the 40-nm-thick ferromagnetic Permalloy (Py) ﬁlm wasevaporated by electron-beam evaporation under the pressure of
2 × 10−7 Pa. About the lateral dimensions of the square dots, weprepared three kinds of samples with different lateral dimensions.
Here, the diagonal distance of the square l varied from 1 μm to
3 μm while the thickness of Py dots in all samples is 40 nm. We
note that the diagonal distance corresponds to the diameter of thecircumscribed circle for the square dot. We conﬁrm that the domain structures at the remanent state for the square dots perfectly
form magnetic vortex structure even for the square dot withl ¼3 μm by using magnetic force microscopy (MFM) (model NanoNavi/E-sweep, SII NanoTechnology) with low moment magnetic
probes. The chirality distribution was random because the nucleation energy for both chiralities is degenerate [23]. The dynamic
properties of the vortex core conﬁned in the square dot have been
evaluated by the reﬂection impedance (S11) measurement using astandard vector network analyzer [12,18]. As shown in Fig. 1(c), a
single Cu strip line with 500 nm in width and 200 nm in thickness
was deposited on the top of the ferromagnetic dots. We note thatthe narrow strip line enables us to detect the resonant oscillationof the magnetization sensitively [18]. The in-plane static magnetic
ﬁeld at an angle ϕ with respect to the strip line was applied toadjust the equilibrium position of the vortex core. Here, we study
the ﬁeld dependence of the resonant property at ϕ ¼0, π /4 and π /2.

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3. Results and discussion
Fig. 2(a) shows S11 signals as a function of the input RF frequency at the remanent state for the three different samples. In allof the samples, the resonant signature was clearly observed as apeak of the signal. The resonant frequencies for the samples withl¼ 1 μm, 2 μm and 3 μm are 405 MHz, 210 MHz and 145 MHz,
respectively. These monotonic changes with the lateral dimension
come from the change of the magneto-static energy and areconsistent with the conventional understandings [21]. In Fig. 2(b),we compare the resonant frequency at the remanent state for thevarious shapes of the ferromagnetic dots [18,24]. Here, the horizontal axis corresponds to the diameter of the circumscribedcircle. The square dot was found to have intermediate values between the circle and the triangle. This indicates that the conﬁned
potential for the vortex core is related to the size of the ferromagnetic dot. So, the magnetic vortex conﬁned in the triangular
dot feels the largest restoring force because the average distancefrom the edges is shortest.
We then study the ﬁeld dependence of the resonant property ofthe magnetic vortex in the square dot as well as its anisotropy. In
the triangular ferromagnetic dot, when the vortex core approaches
to the vertex of the triangle, the resonant frequency increases
because of the enhancement of the conﬁned potential [18]. A similar modulation is also expected in the square-shaped ferromagnetic dot by moving the core around the corner of the squarebecause the vortex structures conﬁned in the corners for thesquare and triangle are resemble each other, as shown in Fig. 3(a).
Moreover, as shown in Fig. 3 (b), when the core moves toward theside of square, the vortex behaves similar to the triangular dot.
Therefore, we expect an insensitive change of the resonant frequency by the application of the magnetic ﬁeld along the side.
First we apply the magnetic ﬁeld perpendicular to the Cu strip(ϕ ¼ π/2), which is a diagonal direction for the square. In this case,the equilibrium core position shifts along the Cu strip. As can beseen in Figs. 4(a) and 4(b), the resonant frequency increases withincreasing the magnitude of the magnetic ﬁeld for both polarities.
This feature is consistent with our expectation that the conﬁned
potential is enhanced when the core moves toward the corner. On
the other hand, as shown in Figs. 4(c) and 4(d), when the magnetic
ﬁeld is applied along the side of the square (ϕ ¼ π/4), the change ofthe resonant property is negligibly small. This in-sensitive behavior is also consistent with the above scenario that the effective
conﬁned potential for the vortex core is not modiﬁed when thecore moves toward the side of the ferromagnetic dot [18]. Thus, inthe square-shaped ferromagnetic dot, the ﬁeld dependence of theresonant frequency strongly depends on the direction of theFig. 1. (a) Spatial distribution of the spin structure in the squared Permalloy dot observed by magnetic-force microscope; (b) numerically simulated vortex structures
stabilized in squared dot under the external ﬁeld parallel to the edge; (c) SEM image of part of the fabricated device; (d) schematic illustration of the sample structure
together with the SEM image of the fabricated sample.

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Fig. 2. (a) S11 spectra as a function of the input RF frequency for the square-shaped ferromagnetic dot with the diagonal distance l ¼1 μm, 2 μm and 3 μm, at the remanentstate; (b) comparison of the resonant frequency of the magnetic vortex conﬁned in circular, square and triangular ferromagnetic dot as a function the diameter of thecircumscribed circle.

Fig. 3. Magnetic vortex structures (a) trapped in the corner of the triangle andsquare and (b) trapped around the side edges.

magnetic ﬁeld. We want to emphasize that the directional modulation of the resonant property in the square dot is larger than
the triangular dot. If we focus on the resonant frequency at H¼200
Oe in Fig. 4, the resonant frequency at ϕ ¼ π/2 is approximately
300 MHz, which is almost twice of that at ϕ ¼ π/4. Thus, a large
modulation of the resonant frequency has been obtained by π/4
rotation of the magnetic ﬁeld. These unique properties of thevortex conﬁned in the square dot are useful for detecting not only
the magnitude of the vortex but also the direction of the magneticﬁeld.
We then study the ﬁeld dependence of the resonant property ofthe magnetic vortex in the square dot as well as its anisotropy.
First we apply the magnetic ﬁeld perpendicular to the Cu strip(ϕ ¼ π/2), which is a diagonal direction for the square. In this case,the equilibrium core position shifts along the Cu strip. As can beseen in Fig. 4(a), the resonant frequency increases with increasing
the magnitude of the magnetic ﬁeld for both polarities. We notethat a similar modulation feature has been observed in a magnetic
vortex conﬁned in the regular triangle. In the triangular ferromagnetic dot, when the vortex core approaches to the vertex of
the triangle, the resonant frequency increases because of theenhancement of the conﬁned potential [18]. Since the similar situation is also expected in the square-shaped ferromagnetic dot bymoving the core around the corner of the square, the modulationof the resonant frequency observed in Fig. 4(a) is caused by theenhancement of the conﬁned potential. On the other hand, as
shown in Fig. 4(b), when the magnetic ﬁeld is applied along theside of the square (ϕ ¼ π/4), the change of the resonant property
is negligibly small. We also want to emphasize that the similar
in-sensitive behavior was obtained in the triangular dot when
the core moves toward the side of the triangle [18]. This is exactly
the same situation in the case of Fig. 3(b). Therefore, the effective
conﬁned potential for the vortex core is not modiﬁed when thecore moves towards the side of the ferromagnetic dot. Thus, inthe square-shaped ferromagnetic dot, the ﬁeld dependence ofthe resonant frequency strongly depends on the direction of themagnetic ﬁeld.
As mentioned above, in the square dot, the resonant frequency
is signiﬁcantly modulated by the application of the magnetic ﬁeld
along the diagonal direction. Here, we discuss about the size dependence of the modulation ratio of the resonant frequency. Fig. 5(a) shows the image plot of S11 spectra as a function of the magnetic ﬁeld along the diagonal direction (ϕ ¼ π/2) for the samples
with l ¼1 μm, 2 μm and 3 μm. In all of the samples, the resonant
frequency increases with increasing the magnitude of the magnetic ﬁeld. We now focus on the modulation ratio η deﬁned by the
maximum resonant frequency divided by the resonant frequency
at H¼0. Fig. 5(b) shows the modulation ratio η as a function of the
diagonal distance l. It can be clearly seen that η monotonicallyincreases with increasing the size of the square. We want to emphasize that the difference in the maximum resonant frequency issmaller than the difference in the minimum resonant frequency,
which corresponds to the resonance at the remanent state. Thismeans that the conﬁned potential for the magnetic vortex does
not depend on the lateral dimension so much when the core islocated in the vicinity of the corner. This can be understood by thefact that one square is perfectly ﬁt at the corner of the othersquare.
Finally, we point out the importance of the relationship between the device conﬁguration and the direction of the core displacement. Fig. 6(a) and (b) shows the image plots for the ﬁeld
dependence of S11 spectra for the square with the diameter of

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Fig. 4. (a) Image plot of S11 signal spectra as a function of external magnetic ﬁeld perpendicular to the strip line (ϕ¼ π/2) and (b) the representative spectra observed atseveral magnetic ﬁelds; (c) image plot of S11 signal spectra as a function of external magnetic ﬁeld along the side of the square ( ϕ = π /4 ) and (d) the representative spectra
observed at several magnetic ﬁelds.

Fig. 5. (a) Image plot of S11 spectra for the square dot with l ¼ 3 μm as a function of the external magnetic ﬁeld along the diagonal direction (ϕ¼π/2); (b) the modulation ratioη as a function of the diameter distance, where the modulation ratio is deﬁned as the ratio of maximum resonant frequency to the resonant frequency at remanent state.

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X. Cui et al. / Physica E 75 (2016) 28–32

in-plane magnetic ﬁeld perpendicular to the strip line. These
modulation properties are useful for detecting the vector information of the magnetic ﬁeld and may provide additional
functionalities in microwave spin devices based on the magnetic
vortex.

Acknowledgments
This work is partially supported by Grant-in-Aid for ScientiﬁcResearch on Innovative Area, “Nano Spin Conversion Science”(26103002) and that for Scientiﬁc Research (S)(25220605). One of
the authors (X.C.) acknowledge the China Scholarship Council for
Young Scientists.

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Fig. 6. Image plots of S11 spectra for the square dot with l ¼3 μm as a function of
external magnetic ﬁeld at (a) ϕ ¼0 and (b) ϕ ¼π/2.

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Therefore, the results indicate that the narrow strip line can detect
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4. Conclusion
We have investigated the resonant property of the magnetic
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