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Size Dependence of Ferromagnetic Resonance Frequency in Submicron Patterned Magnet

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REGULAR PAPER

Japanese Journal of Applied Physics 52 (2013) 053001
http://dx.doi.org/10.7567/JJAP.52.053001

Size Dependence of Ferromagnetic Resonance Frequency in Submicron Patterned Magnet
Takashi Manago1 , Kazuto Yamanoi1 , Satoshi Yakata2;3 , and Takashi Kimura2;3
1

Department of Applied Physics, Fukuoka University, Fukuoka 814-0180, Japan
Advanced Electronics Research Division, INAMORI Frontier Research Center, Kyushu University, Fukuoka 819-0395, Japan
3
CREST, Japan Science and Technology Agency, Chiyoda, Tokyo 102-0075, Japan
E-mail: manago@fukuoka-u.ac.jp
2

Received November 20, 2012; revised March 2, 2013; accepted March 6, 2013; published online April 23, 2013
We investigated the size effect on ferromagnetic resonance (FMR) in a submicron-wide single permalloy bar. The resonant frequency markedly
increased with decreasing bar width to less than 1 m, since the demagnetizing field is effectively modified by changing the bar width even in thinfilms. The resonant frequency difference between 100- and 1000-nm-wide bars was over 4 GHz in the absence of a magnetic field. This
characteristic is promising for practical microwave devices because the desired resonant frequency can be obtained simply by varying the width ofnarrow ferromagnetic bars so that it is not necessary to change the material or magnetic field. # 2013 The Japan Society of Applied Physics

1. Introduction

Spintronic devices operating at high frequencies such as
microwave-assisted magnetization reversal,1,2) spin batteries(spin pumping),3,4) and spin torque diodes,5) rectifying eﬀect
due to time-dependent anisotropic magnetoresistance,6,7)
have been intensively investigated. Additionally, the generation and propagation of spin waves (and magnetostatic
waves) in ferromagnetic metals are also important topics inspin current researches.8–14) These devices are based on theprecessional motion of the magnetization so that ferromagnetic resonance (FMR) is an important phenomenon in them.
The resonant frequency is an important parameter in theiroperation, which is generally determined by the Kittel
equation. We assume that the sample is a thin ﬁlm with abar shape, a static magnetic ﬁeld is applied to the longitudinal
direction of the bar, and an in-plane exciting microwave ﬁeld
is applied perpendicular to the longitudinal direction in plane.
When the crystal anisotropy can be neglected, theresonant frequency can be expressed by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
f ¼
ðB þ ðNy  Nx ÞMS ÞðB þ ðNz  Nx ÞMS Þ ð1aÞ2
 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ð1bÞ
¼
ðB þ Bak ÞðB þ Ba? Þ;2where  is the gyromagnetic ratio, B is the static magnetic
ﬁeld, Nx , Ny , and Nz are demagnetization factors, and MS is
the saturation magnetization. The terms of demagnetization
factors can be described by the anisotropy ﬁelds Bak and
Ba? . Bak is in the ﬁlm plane and Ba? is perpendicular to theplane. For a thin ﬁlm, we usually take Nx ¼ Ny 
¼ 0 andNz ¼ 1, and then Eq. (1) is deduced to a simple form:
 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
BðB þ MS Þ:
f ¼
ð2Þ
2
Equation (2) shows that the resonant frequency depends
only on B and MS , which means that the thin ﬁlm of thesame material has the same resonant frequency at the samemagnetic ﬁeld. It has been presumed that even if Nx 6¼ 0 and
Ny 6¼ 0 in a microfabricated thin ﬁlm magnet, the resonant
frequency is not expected to shift very much because Nx andNy will be small. When many devices are integrated andmicrowaves are irradiated over a wide area, several devices
will be aﬀected by the irradiation. This will cause controllability problems in future applications. In order to control one

particular device, highly localized microwave irradiation isrequired.
Some recent reports, however, implied the possibility of
the resonant frequency shift of FMR in nanoscale magnets.
These reports state the ﬁeld dependence of resonant
frequency with oﬀset even at zero ﬁeld in microwave
assisted magnetic reversal,2) on-chip detection of FMR,15)
and the cone angle measurement of FMR for nanowires.16,17)
The micromagnetic calculation predicted that the resonant
frequency shift occurs in nanosize rectangular ferromagnetic
stripes.18) These reports drop a hint that the resonant
frequency can be tunable in nanoscale magnets even in thesame material because the demagnetization eﬀect cannot be
neglected.
Recently, some trials of vector network analyzer ferromagnetic resonance (VNA-FMR) measurement have beenreported.19–22) In this technique, the sample is mounted on
top of a coplanar waveguide (CPW) structure on a wafer.
Microwaves are injected via the CPW and the S-parameter isdetermined from the microwave transmission or reﬂection
loss. The VNA-FMR has the advantage that local excitationnear the CPW can be realized and it has a higher sensitivity
than conventional FMR. For the submicron magnets, however, the FMR characteristics are measured as the average ofarray of many submicron magnets.2) It includes the static
magnetic interaction between magnets and diﬀerent fromsingle magnet properties.
In this study, we systematically investigated the size
dependences of FMR characteristics in a single submicronbar of permalloy (Py) using a VNA. We found that theresonant frequency can be controlled over a wide range byvarying the bar width on a submicron scale.
2. Experimental Procedure

A CPW structure of Cu and a submicron Py bar were
fabricated on a Si/SiO2 substrate using electron beamlithography, electron beam evaporation, and the lift-oﬀ
technique. Figure 1(a) shows SEM images of the sample.
The CPW structure consists of two planes and is designed tohave an impedance of 50 , which matches that of the VNA
and cables. The CPW is of the signal-ground (SG) type andis terminated by a 2-m-wide strip at the end of the twoplanes. The Py bar of 30 nm thickness and 6 m length was
fabricated on a shorted strip. The Py width was varied
between 100 and 1000 nm.

053001-1# 2013 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 52 (2013) 053001

T. Manago et al.
0.003

(a)

Width 1000 nm

Py Width

100 mT

Bext
brf

0.002

Py wireRe ΔS11

6 μm

Waveguide

2 μm

(b)
Anisotropy field

Ba⊥

10 mT

0 mT

0.001

Ba//

Excitation field brf

CPW
0
2

Bext

FMR measurements were performed using a VNA(HP8510C) and a microprobe station (Cascade Microtech
Summit 9000). The stage of the microprobe station was
modiﬁed to apply an in-plane static magnetic ﬁeld. As
shown in Figs. 1(a) and 1(b), a static magnetic ﬁeld, Bext ,was applied along the longitudinal direction of the bar andan excitation microwave magnetic ﬁeld, brf , was applied
perpendicular to the longitudinal direction of the bar. Bak
and Ba? are anisotropy ﬁelds due to the shape anisotropy ofthe sample. In this conﬁguration, the excitation spin wave of
the Kittel mode (homogeneous rotation mode) is dominant,since the excitation ﬁeld is applied to the whole bar. We
performed one-port reﬂection measurements and determined
the S-parameter (S11 ). The incident microwave power was
0 dBm, which was conﬁrmed under a linear response regime
by the power dependence of the spectra shape. The signals
obtained under a static magnetic ﬁeld were analyzed bysubtraction and phase-correction using the backgroundsignal, which was obtained under zero magnetic ﬁeld.
Hereafter, we called this subtracted signal S11 . The S11
response derived from linearized LLG equation in a small
excitation ﬁeld regime is
i!d ð!20  !2 Þ þ !ð!2d þ !2 Þ;ð!20  !2 Þ2 þ 2 !2 ð!c þ !d Þ2

6

8

10

12

Frequency (GHz)

Fig. 1. (Color online) (a) SEM images of Py bar on CPW structure.
(b) Magnetic ﬁeld directions in present measurement. Bext is the static
magnetic ﬁeld, brf is the excitation microwave magnetic ﬁeld, and Bak andBa? are the anisotropy ﬁelds due to the shape anisotropy of the Py bar.

S11 /

4

ð3Þ

where ! ¼ ðBak þ Bext Þ, !d ¼ ðBa? þ Bext Þ, !0 ¼
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c!c !d ,  is the damping factor, and  ¼ gB =h (B is
the Bohr magneton and g-factor is 2.11, which is a substitutefor the Fe value). The resonant frequency, anisotropy ﬁelds,and  were estimated by ﬁtting the spectra of the real part of
S11 . The Py width dependences of these parameters weresystematically investigated in detail.
3. Results and Discussion

Figure 2 shows the magnetic ﬁeld variation of the Re S11
spectra and their ﬁt curves for the bar with 6  1 m2 . The
spectra contain a resonant peak caused by the inductancechange of the FMR. The resonant frequency increases with a

Fig. 2. (Color online) Magnetic ﬁeld variation of Re S11 spectra andtheir ﬁt curves for bar with 6  1 m2 .

magnetic ﬁeld, as expected from the Kittel equation. Note that
FMR was observed even at 0 mT, which suggests that the termðNy  Nx ÞMS or the anisotropy ﬁeld of the width direction(Bak ) in Eq. (1) cannot be neglected. These spectra were ﬁtted
using the real part of Eq. (3) and the parameters , Bak , andBa? were extracted. The obtained  can include extrinsic
contributions such as two-magnon scattering, which increases
the FMR linewidth. This is caused by inhomogeneities anddefects at surface and interfaces, and transfers energy froman uniform FMR mode to degenerate short-wavelength spinwaves.23–25) Since this contribution is important in ultrathin
ﬁlms, the obtained  does not represent an intrinsic damping
parameter. These are summarized in Figs. 3(a) and 3(b). 
was about 0.02, and Bak and Ba? were about 21 and 916 mT,
respectively. These values should be constant because thesample did not change during the measurement. The small
deviation of the damping factor could be ﬁtting errors becauseof noisy signals due to the very small samples. It is known
that the damping factor of bulk Py is about 0.007. Our otherlarger samples prepared by the sputtering deposition of Py
on a Si substrate, etching to a bar shape, and CPW depositionon the Py bar, showed an  of about 0.01, which is usuallyobtained for ﬁlm samples. In the present case, the Py bar
was made on a CPW of Cu. The grains of Cu are large.
Therefore, we think that the large  was aﬀected by theinhomogeneiteis and roughness of the Py bar.
Figure 4 shows the bar width variation of the Re S11
spectra and their ﬁt curves. The resonant frequency
markedly increases with decreasing bar width. The peakmagnitude decreases since the microwave energy loss is
expected to be proportional to the volume of the magnet. In
Fig. 5(a), the bar width variation of  except of 100 nm
hardly changed, which is reasonable because these samples
were prepared at the same time and thus the sample
conditions were the same. The  of the 100-nm-widesample, however, becomes large. We think that the edge
roughness in the very narrow bar also largely aﬀects thedamping factor. Figures 5(b) and 5(c) show the variation of
the anisotropy ﬁelds with the bar width and the relationshipbetween the resonant frequency and the bar width. In

053001-2# 2013 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 52 (2013) 053001

T. Manago et al.

0.05

Damping factor α

Damping factor α

0.04
0.03
0.02
0.01
0

0.01

Banisotropy (mT)

100

Ba//
Ba//
20

0.02

0

Ba⊥

0

0.03

40
60
80
Magnetic field (mT)

100(b) Magnetic ﬁeld dependence of anisotropy ﬁelds Bak , Ba? .

0.003
100 nm1000 nm

Re ΔS11

0.002

10

11

12

13

Ba⊥

850

Ba//

150

Ba//

100

0
14

(c)

12
100 mT

10

60 mT

8

30 mT

6
Length: 6 μm
Thickness: 30 nm

4
2

0

200

0 mT

400 600 800
Py width (nm)1000 1200

Fig. 5. (Color online) (a) Variation of damping factor  with Py width.
(b) Variations in anisotropy ﬁelds Bak and Ba? with Py width. (c) Bar width
dependence of resonant frequency at static magnetic ﬁelds of 0, 30, 60, and100 mT.

Frequency (GHz)

300 nm
0.001

Ba⊥

900

50

120

Fig. 3. (Color online) (a) Magnetic ﬁeld dependence of damping factor .

500 nm 9

Length 6 μm
Thickness 30 nm

(b)

950

900

0

0.04

Resonant frequency (GHz)

Banisotropy (mT)

Ba⊥

50

0.05

6 x 1 μm
Thickness 30 nm

(b)
950

(a)

0.06

(a)

200 nm
100 nm

0
6

7

8

9

10

11

12

13

Frequency (GHz)
Fig. 4. (Color online) Bar width variation of Re S11 spectra and their ﬁtcurves with widths of 100–1000 nm.

Fig. 5(b), Bak increases and Ba? decreases slightly as thebar width decreases below 500 nm. Accompanying thesechanges in the anisotropy ﬁelds, the resonant frequency
increases markedly as the bar width decreases below 500
nm [Fig. 5(c)]. The resonant frequency increases by about
2 GHz at 100 mT and more than 4 GHz at 0 mT as thebar width is decreased from 1000 to 100 nm. For the bar
narrower than 100 nm, a greater frequency shift is expected.
This frequency shift increases as the static ﬁeld is reduced.
Thus, the change in the anisotropy ﬁelds is not so large, but
the eﬀect on the resonant frequency is large for bars withsubmicron widths.
Figure 6(a) shows the static magnetic ﬁeld dependenceof the resonant frequency for each width sample. The solidline is a calculated curve using the Kittel equation for a thinﬁlm [Eq. (2)]. The resonant frequency of the narrow bars is

shifted upward from the solid curve. This frequency shift
increases with decreasing bar width. Additionally, there is aresonant peak at 0 T and the frequency shift is largest at
0 T compared with the case under ﬁnite ﬁelds. Thus, thedemagnetization eﬀect of thin ﬁlms for the ferromagnetic
resonance becomes large when the sample width is less than
1 m. The resonant frequency can be largely controlled bychanging the sample width for narrow bars. It is worthwhile
to investigate the classical demagnetization factor. The
demagnetization factors were estimated by curve ﬁtting
using the general Kittel equation [Eq. (1b)]. MS is 1.08 T
and the ﬁtting parameters are Ny and Nz , and Nx is obtainedby Nx ¼ 1  Nz  Ny . The dashed lines in Fig. 6(a) arecurves obtained by this ﬁtting. Figure 6(b) shows therelationship between the demagnetization factors and thePy width. Ny increases and Nz decreases slightly as the bar
width decreases below 500 nm. The estimated anisotropy
ﬁelds using these demagnetization factors are consistent withthe anisotropy ﬁeld obtained by spectral ﬁtting. We compared the result with the analytically calculated demagnetization factor Nx for rectangular prisms using Aharoni’s
equation.26) The range of these values is 0.0040 for the
100-nm-wide sample to 0.0074 for the 1000-nm-widesample. In the ﬁtting results (Nx ¼ 0:04{0:05), there isa weak sample width dependence, and what is much

053001-3# 2013 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 52 (2013) 053001

T. Manago et al.

irradiated over a wide area, a certain dot or bar can beselected. One additional advantage of a narrow bar is that
resonance occurs at 0 mT so that it is not necessary to apply
a static magnetic ﬁeld, and the resonant frequency can be
widely changed at a zero magnetic ﬁeld than at a ﬁnite ﬁeld.
This is also very useful for spin pumping because a static
magnetic ﬁeld is not required to generate a spin current.
When there are several bars with diﬀerent widths, thelocation at which the spin current is generated can also becontrolled.

14(a)Resonant frequency (GHz)

12
10
8
6
Width

100 nm
200 nm
300 nm
500 nm1000 nmKittel eq. for thin film

4
2
0
-40

-20

0

20

40

60

80

Acknowledgments

This work was partly supported by the Japan Society for
the Promotion of Science (JSPS), Grant-in-Aid for ScientiﬁcResearch (C: 24560034). This work was partly supported
by funds (No. 115002) form the Central Research Instituteof Fukuoka University. The authors thank Dr. Kasai of the
National Institute for Materials Science (NIMS) for fruitful
discussions.

100

Magnetic field (mT)

1
Demagnetization factor

(b)
0.9

z
0.8

y
x
0.2

1) C. Thirion, W. Wernsdorfer, and D. Mailly: Nat. Mater. 2 (2003) 524.
2) Y. Nozaki, M. Ohta, S. Taharazako, K. Tateishi, S. Yoshimura, and K.

Nx
Ny
Nz

Matsuyama: Appl. Phys. Lett. 91 (2007) 082510.
3) A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I. Halperin: Phys. Rev.

B 66 (2002) 060404(R).

0.1

4) S. Mizukami, Y. Ando, and T. Miyazaki: Phys. Rev. B 66 (2002) 104413.
5) A. A. Tulapurkar, Y. Suzuki, F. Fukushima, H. Kubota, H. Maekawa, K.

0
0

200

400 600 800
Py width (nm)

23)
24)
25)
26)

Tsunekawa, D. D. Djayaprawira, N. Watanabe, and S. Yuasa: Nature 438(2005) 339.
A. Yamaguchi, K. Motoi, A. Hirohata, H. Miyajima, Y. Miyazaki, and Y.
Sanada: Phys. Rev. B 78 (2008) 104401.
A. Yamaguchi, K. Motoi, A. Hirohata, and H. Miyajima: J. Appl. Phys.
105 (2009) 07D301.
K. Y. Guslienko, S. O. Demokritov, B. Hillbrands, and A. N. Slavin: Phys.
Rev. B 66 (2002) 132402.
M. Bailleul, D. Olligs, C. Fermon, and S. O. Demokritov: Europhys. Lett.
56 (2001) 741.
Y. Roussigné, S. M. Chérif, C. Dugautier, and P. Moch: Phys. Rev. B 63
(2001) 134429.
M. Covington, T. M. Crawford, and G. J. Parker: Phys. Rev. Lett. 89(2002) 237202.
Y. S. Gui, N. Mecking, and C.-M. Hu: Phys. Rev. Lett. 98 (2007) 217603.
V. Vlaminck and M. Bailleul: Science 322 (2008) 410.
K. Sekiguchi, K. Yamada, S.-M. Seo, K.-J. Lee, D. Chiba, K. Kobayashi,
and T. Ono: Phys. Rev. Lett. 108 (2012) 017203.
M. V. Costache, M. Sladkov, C. H. van der Wal, and B. J. van Wees: Appl.
Phys. Lett. 89 (2006) 192506.
M. V. Costache, S. M. Watts, M. Sladkov, C. H. van der Wal, and B. J. van
Wees: Appl. Phys. Lett. 89 (2006) 232115.
N. Kuhlmann, A. Vogel, and G. Meier: Phys. Rev. B 85 (2012) 014410.
O. Gérardin, H. Le Gall, M. J. Donahue, and N. Vukadinovic: J. Appl.
Phys. 89 (2001) 7012.
G. Counil, J.-V. Kim, T. Devolder, C. Chapper, K. Shigeto, and Y. Otani:
J. Appl. Phys. 95 (2004) 5646.
Y. Ding, T. J. Klemmer, and T. M. Crawford: J. Appl. Phys. 96 (2004)
2969.
B. K. Kuanr, R. E. Camley, and Z. Celinski: J. Magn. Magn. Mater. 286(2005) 276.
S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P.
Kabos, T. J. Silva, and J. P. Nibarger: J. Appl. Phys. 99 (2006) 093909.
R. Arias and D. L. Mills: Phys. Rev. B 60 (1999) 7395.
M. J. Hurben and C. E. Patton: J. Appl. Phys. 83 (1998) 4344.
M. Sparks, R. London, and C. Kittel: Phys. Rev. 122 (1961) 791.
A. Aharoni: J. Appl. Phys. 83 (1998) 3432.

053001-4# 2013 The Japan Society of Applied Physics

1000 1200
6)

Fig. 6. (Color online) (a) Static magnetic ﬁeld dependence of resonant
frequency for bars with widths of 100–1000 nm. The dashed lines are curves
obtained using the general Kittel equation [Eq. (1)], while the solid line isthe curve calculated using the Kittel equation for a thin ﬁlm [Eq. (2)].
(b) Variation in demagnetization factors with Py width.

7)
8)
9)
10)

larger than the calculated results. This discrepancy is
attributable to the disturbance of the magnetic moments
around the edge and roughness of the bar, which could
locally enhance the actual demagnetization factor in thelongitudinal direction.

11)
12)
13)
14)
15)

4. Conclusion

We investigated the FMR characteristics of single submicronbars of permalloy (Py) using a VNA and found that theresonant frequency can be controlled over a wide range byvarying the bar width on a submicron scale. This originatesfrom the anisotropy ﬁeld due to the shape magnetic
anisotropy. In a nanoscale bar, the demagnetization ﬁeld in
the plane cannot be neglected even in the thin ﬁlm because
the thickness and ﬁlm width could be comparable. These
results provide guidelines for designing microwave devices.
Even when the same magnetic material is used for
microwave applications, the resonant frequencies of individual submicron magnets on a single chip can be tuned
by varying the width of the magnets. If microwaves are

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