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OPEN

received: 31 March 2015
accepted: 13 October 2015
Published: 09 December 2015

Sensitive detection of vortexcore resonance using amplitudemodulated magnetic field
Xiaomin Cui1, Shaojie Hu1, Makoto Hidegara2, Satoshi Yakata3 & Takashi Kimura2,4,5Understanding and manipulating the dynamic properties of the magnetic vortices stabilized inpatterned ferromagnetic structures are of great interest owing to the superior resonant features withthe high thermal stability and their flexible tunability. So far, numerous methods for investigating the
dynamic properties of the magnetic vortex have been proposed and demonstrated. However, thosetechniques have some regulations such as spatial resolution, experimental facility and sensitivity. Here,
we develop a simple and sensitive method for investigating the vortex-core dynamics by using theelectrically separated excitation and detection circuits. We demonstrate that the resonant oscillation of
the magnetic vortex induced by the amplitude- modulated alternating-sign magnetic field is efficiently
picked up by the lock-in detection with the modulated frequency. By extending this method, we also
investigate the size dependence and the influence of the magneto-static interaction in the resonant
property of the magnetic vortex.
Ferromagnetic materials with reduced dimensions show unique arrangements of magnetic spins by introducing thegeometrically controlled magneto-static interactions. Such patterned domain structures are useful for applicationin future spintronic devices as well as for understanding of the fundamental spin-related physics1,2. Especially, themagnetic vortex structure, stabilized in a ferromagnetic disk with a diameter less than a micron, has a potential
as a unit cell for high density magnetic storage because of negligible magnetostatic interaction2–6. In addition, thespin dynamics of magnetic vortices stabilized in micron or submicron scaled ferromagnetic disks are of particularinterest because the wide range tuning of the eigenfrequency for the vortex can be realized by its sample dimension7–10. Moreover, its low frequency dispersion at the resonant oscillation with a high thermal stability is anothergreat performance of the magnetic vortex11. Therefore, understanding the spin dynamics of the magnetic vortex
is an important issue for further developments of the spin-related physics and its application.
So far, numerous methods for investigating the dynamic properties of the magnetic vortex have been proposedand demonstrated. Time resolved magneto-optical Kerr effect is a powerful technique to study ultrafast spin
dynamics in ferromagnetic films and their patterned structures12,13. However, the spatial resolution is limited by
the wave length of the laser beam, typically sub-micron range. Instead of the laser beam, soft X-ray microscopy
enables us to image the detailed spin structures with sub-nanosecond time and nano-meter spatial resolutions14.
However, this kind of experiment is only carried out under the limited conditions because of the large scale facility. Apart from the direct imaging technique, the detection of the magnetization dynamics using the electricaltransport measurements are widely utilized. High frequency impedance measurements of a coplanar wave guideor microstrip line with nanomagnets using network analyzer enable to detect the translational resonant motionsof the magnetic vortices15–17. However, because of the sensitivity of the inductive coupling, approximately 1000
nanomagnets are required for detecting the vortex dynamics. Homodyne detection technique using the magnetoresistance effects sensitively detects the dynamical vortex motion of a single ferromagnetic dot18–20. But its
analysis is complicated because of the coexistence between the Oersted field and spin-transfer torque. In addition,
the monolithic radio-frequency (RF) measurement schemes used in the homodyne electrical techniques regulate
the situation of vortex excitation and prevent the detailed study of the dynamic characteristic. Sugimoto et al.
1

Graduate School of Information Science and Electrical Engineering, Kyushu University, 744 Motooka, Fukuoka, 8190395, Japan. 2Department of Physics, Kyushu University, 6-10-1 Hakozaki, Fukuoka 812-8581, Japan. 3Department
of Information Electronics, Fukuoka Institute of Technology, 3-30-1 Wajiro-higashi, Higashi-ku, Fukuoka, 811-0295
Japan. 4Research Center for Quantum Nano-Spin Sciences, Kyushu University, 6-10-1 Hakozaki, Fukuoka 812-8581,
Japan. 5CREST, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075, Japan. Correspondence andrequests for materials should be addressed to T.K. (email: t-kimu@phys.kyushu-u.ac.jp)
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Figure 1. (a) Device configuration together with a SEM image of part of the device; (b) Schematic illustrationof the proposed measurement setup for the vortex dynamics. During the measurement, an in-plane bias static
magnetic field is applied parallel with the chain of Py disks. The amplitude modulated RF field is injected byflowing on the circuit of periodically patterned Cu electrode which locates on the top of the Py disk with aninsulation layer of SiO2. The voltage change is monitored by another separated circuit combining the lock-in
measurement system.

developed another homodyne detection technique using two independent RF currents flowing in the excitationand detection circuits21. However, the RF current flowing in the detecting circuit may affect the vortex dynamics
because the frequency of the detecting current is close to the resonant frequency. Here, by using an amplitude
modulated RF magnetic field for exciting the vortex resonance, we develop a sensitive lock-in detection method
for the vortex dynamics.
Figure 1 shows the schematic device structure for our measurement setup together with a scanning electron
microscope image of part of the patterned circular disks. The sample consists of a chain of 40-nm-thick Permalloy(Py) dots connected by 200-nm-thick Cu pads and a periodical Cu electrode on the Py disks. Here, the numberof the dot is 50, much smaller than that for network analyzer measurement16. The electrical connection between
the Py dot and Cu electrode is insulated by a patterned SiO2 film whose thickness is 100 nm. The Py disks withdiameters varying from 2 μm to 4 μm were fabricated by using a conventional lift-off method combined with theelectron beam lithography. The Cu electrodes were deposited by a Joule-heat evaporator after the surface cleaningof the Py disk by a very low energy Ar ion milling. The electrical resistivity for the Py is 30 μΩ cm, much larger
than that for the Cu electrode (2.0 μΩ cm)22. The anisotropic magnetoresitance (AMR) measurements under themagnetic field with the various direction were performed by two-terminal resistance measurement of 10 μA. For
the vortex dynamics measurement, the static magnetic field is applied along the horizontal (x) direction wherethe vortex core is shifted along y direction23. The resonant motions of the vortices are excited by injecting the RF
signal into the Cu periodical electrode. The dynamic response of the vortices under the RF magnetic field withsweeping the frequency is monitored by measuring the voltage of the chain of the disk through the AMR change,
as explained below.
Figure 2(a) shows the longitudinal and transverse AMR curves for the sample with the diameter of 2 μm. In
the measurements, the magnetic field does not exceed the annihilation field 20 mT, meaning that the vortex coreis located in the disk during the measurement. As shown in Fig. 2(a), the field-dependence of the resistance is wellfitted by the parabolic dependence. Here, the parabolic curve in the longitudinal AMR shows the horizontal shift.
This is because the Cu electrode has a vertical offset from the center of the disk. By changing the direction of themagnetic field with assuming the linear field dependence of the core position under the low magnetic field6,17,24,
we obtain the core-position dependence of the resistance as shown in Fig. 2(b).
To mathematically describe the position dependence of the resistance, we focus on the two-fold rotational
symmetry originating from the anisotropic magnetoresistance. Besides, the resistance change due to the vortex
core displacement along the x direction should be smaller than that along the y direction because the core displacement along the x axis is partially smeared out by the Cu pads. These features are surely confirmed in theexperimental results. From these considerations with parabolic dependence of the resistance change on the core
position shift, we found that the resistance change (from the origin (0, 0)) ∆R (x , y) as a function of the twodimensional core position (x , y) can be given by the following equation.
2

2
2
 x   (y − y off )x  −   ∆R (x , y) = ∆AMR  1 − α   2 r   r  
 
r



(1)

Here, ∆AMR is the AMR change induced by the core displacement from the center to the position under magnetic
field 10 mT along the x axis. r is the displacement of the vortex core due to the application of the magnetic field
10 mT along the x axis and α is the correction factor introduced by taking account of the influence of the Cu

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Figure 2. (a) Longitudinal and transverse AMR curves for the chain of the Py disks with 2 micron diameter.
The AMR was measured using the traditional probe configuration for both H ⊥ I and H I , respectively.
(b) Experimentally obtained resistance change (normalized) as a function of the core position. The AMR curveof the device was detected with changing magnetic field directions. Since the magnitude of the magnetic field issmaller than the annihilation field, we assume a linear field dependence of the core position. (c) Calculated
position dependence of the resistance change from the origin (0, 0) based on Eq. (1) with α =  0.05.

electrode for flowing dc current in the Py. We note that this description can be derived from the bivariate polynomial approximation. When the core is shifted to x direction from the center, the asymmetry with respect to theelectrode increases. In such a situation, since the average y component of the magnetization becomes non zero,
the resistance change due to the core displacement along y axis decreases compared to that at x =  0. Thus, theposition dependence of the resistance is affected by the electrode. By fitting the experimental result shown in
Fig. 2(b) to Eq. (1), the correction factor alpha for the present device is found to be 0.05. yoff is the offset due to thevertical shift of the Cu electrode. As seen in Fig. 2(c), the equation well reproduces the experimental results shownin Fig. 2(b).
We then consider the average resistance R (x , y) under the amplitude-modulated RF magnetic field with thestatic magnetic field along the x axis. The trajectory of the vortex core during resonant oscillation state induced by
the RF magnetic field is known to be the circular shape not only in the absence of the magnetic field but also in
the in-plane static magnetic field sufficiently smaller than the annihilation field7,17,25. Therefore, we adapt thefollowing equations as the core trajectory under the amplitude-modulated RF magnetic field.
x = δ (1 + m cos ω mt) cos ωRFt ,y = y 0 + δ (1 + m cos ω mt) sin ωRFt

(2)Here, δ is the oscillation radius of the vortex core and its frequency dependence takes the Lorentzian resonant line
shape. m is the modulation ratio and y 0 is the vertical shift of the core position due to the application of the horizontal static magnetic field. ω m and ωRF are the modulation and RF frequencies, respectively.
The induced voltage can be calculated by substituting Eq. (2) for Eq. (1) with multiplying I dc. By using lock-in
detection technique with ω m, we can pick up only the coefficient for sin ω mt in the ac voltage. Since the experimentally detected voltage V ω m in lock-in amplifier is the time average, we obtain the following relationship for theeffective resistance R between oscillation state and non-oscillation state (where δ is 0), which is defined by theaverage ac voltage 〈V ω m〉 divided by I dc.
R≡〈V ω m〉I dc= ∆AMR222
 y − y off  mα  δ    δ       + 2  0



 
2  r    r r(3)Here, The 1st term depends on the frequency but does not depend on the vertical shift y 0. The 2nd term depends
both on the frequency and the vertical shift. In the comparison between the 1st and 2nd terms, δ/r , which is therelative oscillation amplitude, depends on the input RF power, but, in general, is smaller than 0.1 even at the resonant state7. However, (y 0 − y off )/ r easily exceeds 0.1 by the application of the horizontal static magnetic field.
Therefore, the 2nd term is dominant in Eq. (3). We emphasize that the detected voltage does not include any
background signal, which is independent of the oscillation. This indicates that the present modulation method
sensitively detects the voltage change due to the vortex oscillation.
To demonstrate the proposed technique for detecting the core resonance, we measure the effective resistance〈V ω m〉/ I dc under the RF magnetic field. First, we measured the bias current dependence of the voltage spectra
under the static magnetic field of 5 mT (Fig. 3(b)). Here, the dc current is varied from 1 mA to 20 mA. As can beseen in Fig. 3(b), we do not see any significant change of resonant frequency in this current range. The result
indicates that the influence of the dc current on the vortex core resonance is negligibly small below 20 mA. This isconsistent with the previous theoretical and experimental study24,26. From this consideration, we decided to use
6 mA, which is sufficiently small, in the series of the measurements presented below. Figure 3(c) shows the effective
resistance as a function of the RF frequency at the fixed external fields of 0, 3.2 mT and 6.4 mT. Clear resistancedips ∆Rres due to the vortex-core resonance are observed. Here, the magnitude of the dip ∆Rres in the spectra is

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Figure 3. (a) Two experimental parameters; DC current I dc for generating the voltage including the resonant
signature and static magnetic field H parallel with the dc current (x axis) for moving the core position along yaxis. (b) Current dependence of the voltage spectra for the chained disk with the disk diameter of 2 μm under
the static magnetic field of 5 mT; (c) Frequency dependence of the average resistance change 〈V ω m〉/ I dc for the
disk with 2 micron diameter for various static magnetic field. The voltage spectra were detected by flowing DC
current of 6 mA with sweeping the RF frequency. (d) Field dependence of the resistance change ∆Rres due tovortex core resonance. ∆Rres is defined as the resistance change between base line and resonance dip. The data is
well fitted by a parabolic curve (solid line) based on Eq. (3). (e) The resonant frequency of the disk with
2 micron diameter as a function of bias static field. The frequency shows a weak dependence on the static
magnetic field.

confirmed to increase with the static field. Moreover, as shown in Fig. 3(d), the field dependence of ∆Rres is wellfitted by the parabolic equation. From the fitting, we find that H off takes a positive value. This means that the
number of the disks with CW chirality is larger than that with CCW chirality in this measurement. These experimental facts are highly consistent with the above expectation. We also point out that the resonant frequency isalmost independent of the static magnetic field because the core potential is well expressed by the parabolic
potential. This is clear evidence that a circular ferromagnetic disk creates a well-defined harmonic potential17,27.
Moreover, we also plotted the resonant frequency as a function of the static magnetic field as indicated in Fig. 3(e).
The resonant field shows the weak field dependence although the signature expected from the ideal parabolic
potential should not depend on the magnetic field. The origin of this weak dependence may be related to nonlinear
behavior of the resonant frequency and/or the formation of the anharmonic magnetostatic potential in the circulardevices under a bias static field28.
We also perform the similar measurements for the chain of the disk with the different diameter. Figure 4(a)shows the spectra of the effective resistance 〈V ω m〉/ I dc at μ0H =  − 4 mT for various disk diameters, 2 μm, 3 μm and4 μm. Here, the annihilation fields of the vortex for 2 μm-, 3 μm- and 4 μm-diameter disks are about 20 mT, 18 mT
and 16 mT, much larger than 4 mT. In each spectrum, a resistance dip due to the vortex resonance is clearlyobserved. The resonant frequencies for 2 μm-, 3 μm- and 4 μm-diameter disks are 180 MHz, 135 MHz and 105 MHz,
respectively, showing the monotonic increase with reducing the diameter of the disk. The observed size dependenceof the resonant frequency is well reproduced by the numerical simulation using the object-oriented micromagnetic
frame network (OOMMF). Numerically obtained resonant frequencies for 2 μm, 3 μm, and 4 μm disks are, ,respectively, 177 MHz, 132 MHz and 86 MHz, in consistent with the experimental results. Besides, we study thevortex dynamics for the magneto-statically coupled vortex system. As shown in the inset of Fig. 4(b), we havefabricated the chain of the magneto-statically coupled disks. Here, the disk diameter is 3 μm and the edge-to-edge
distance between the neighboring disk is 300 nm. Figure 4(b) shows the spectrum of the effective resistance〈V ω m〉/ I dc for the chain of the coupled vortices. The resistance dip due to the vortex resonance is clearly observed
at f = 120 MHz, which is lower than that for the isolated disk with 3 μm diameter shown in Fig. 3(c), suggesting
an in-phase oscillation mode of the coupled vortices29–31. This excitation mode can be extended from the resonance

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Figure 4. (a) 〈V ω m〉/ I dc spectra at H =  − 4 mT for various disks with diameters of 2 μm, 3 μm and 4 μm. The
measurements have been carried out under the DC current of 6 mA and the RF amplitude of 5 dBm.
(b) 〈V ω m〉/ I dc for the magneto-statically couple vortices with the diameter of 3 μm . The inset shows a SEM
image of the fabricated device where the edge-to-edge interval is 300 nm.

in the pair of the coupled vortex dynamics with in-phase oscillation to a chain, which has been confirmed numerically and experimentally8,29,32.
We have developed a sensitive detection technique of the magnetic vortex resonance. Electrically separated
excitation and detection circuits enable us to pick up the vortex oscillation without any background signal, resulting in the sensitive detection of the vortex resonance. The magnetic-field dependence of the resonant signal iswell explained by the simple analytical model without considering spurious effects such as Oesrted field andspin-transfer torque. The obtained size and interval dependences of the resonant property show good agreementwith the previously reported numerical and experimental results7,18. By extending the present technique to theapplication of the spatially modulated magnetic field, it is possible to excite the unique high-energy resonant
mode8,26.

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Acknowledgements

This work is partially supported by Grant-in-Aid for Scientific Research on Innovative Area, “Nano Spin Conversion
Science” (26103002) and that for Scientific Research (S) (25220605). The authors (Xiaomin Cui and Shaojie Hu)
acknowledge the China Scholarship Council for Young Scientists and Marubun Foundation.

Author Contributions

X.C. carried out the main part of the experimental work, including the preparation of the sample and wrote
the main manuscript text including the figure preparations. S.H. prepared the sample and M.H. carried out thenumerical simulation. S.Y. and T.K. supervised the experimental research and finalized the manuscript. All authors
reviewed the manuscript.

Additional Information

Competing financial interests: The authors declare no competing financial interests.
How to cite this article: Cui, X. et al. Sensitive detection of vortex-core resonance using amplitude-modulated
magnetic field. Sci. Rep. 5, 17922; doi: 10.1038/srep17922 (2015).
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