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JOURNAL OF APPLIED PHYSICS 115, 053914 (2014)
Propagation of nonlinearly generated harmonic spin waves
in microscopic stripes
O. Rousseau,1 M. Yamada,2 K. Miura,2 S. Ogawa,2 and Y. Otani1,3,a)
1
Center for Emergent Matter Science, RIKEN, 2-1 Hirosawa, Wako 351-0198, Japan
Hitachi, Ltd., Central Research Laboratory, 1-280 Higashi-koigakubo, Kokubunji, Tokyo 185-8601, Japan
3
Institute for Solid State Physics, University of Tokyo, Kashiwa 277-858, Japan
2
(Received 25 September 2013; accepted 24 January 2014; published online 7 February 2014)
We report on the experimental study of the propagation of nonlinearly generated harmonic spin
waves in microscopic CoFeB stripes. Using an all electrical technique with coplanar waveguides,
we find that two kinds of spin waves can be generated by nonlinear frequency multiplication. One
has a non-uniform spatial geometry and thus requires appropriate detector geometry to be
identified. The other corresponds to the resonant fundamental propagative spin waves and can be
efficiently excited by double- or triple-frequency harmonics with any geometry. Nonlinear excited
spin waves are particularly efficient in providing an electrical signal arising from spin wave
C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4864480]
propagation. V
In recent years, spin waves (SWs) have been studied for
implementation in logic circuits.1–7 The use of metallic ferromagnetic films in such devices is expected to provide better integration with the already existing transistor technology
on silicon than dielectric ferromagnetic devices such as yttrium iron garnet. This is not only through the possibility of
patterning offered by magnetic metallic films but also having
better electrical control over the dynamic magnetization. The
propagation of SWs and their properties such as velocity,
direction or geometry, in various devices7–10 has already
been investigated using metallic ferromagnetic films with all
electrical inductive measurements. However, in these measurements, the electrical signal obtained after propagation of
the SW is mainly the result of couplings between detection
and excitation areas. It then seems difficult to use this type of
SW signal for logic. Dynamic magnetization is well
described by the Landau-Lipshitz equation, which is intrinsically a nonlinear equation. Nonlinear effects can be very useful as some new frequencies in the dynamics can be
generated by a combination of frequencies already existing
in the magnetic system.11–16 Indeed, if some harmonic SWs
are nonlinearly generated, then there will be no other signal
at this harmonic frequency and the SW signal can be used
for logic. We report here on the propagation of nonlinearly
generated harmonic SWs using an entirely electrical device.
The geometry of our samples is shown in Figure 1(a).
The samples consist of a Ta/Ru/Ta/CoFeB (20 or 10
nm)/MgO (CFB) film on a Si/SiO2 substrate patterned by electron beam lithography and ion etching into stripes of either
6 lm or 1.5 lm wide. For excitation and detection, two coplanar waveguides (CPWs)7,9,10 are used. The principle of the
experiment is sketched in Figure 1(b). In the present work, a
continuous radio frequency (rf) current is passed through
CPW1 to generate a rf magnetic field. This rf magnetic field
generates dynamic magnetization in a long ferromagnetic
stripe, especially SWs at the resonant condition for magnetic
a)
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-8979/2014/115(5)/053914/5/$30.00
field and frequency of excitation. After propagation along the
stripe, the resulting dynamic magnetization s is detected
inductively by means of CPW2. The CPWs are made by electron beam lithography and electron beam evaporation of Ti(5
nm)Au(150 nm). The center-to-center distance between
CPWs is 8, 9, 10, 11, or 12 lm depending on the sample. To
FIG. 1. (a) SEM image of a device. (b) Sketch of the operating principle for
the nonlinear excitation. (c) Normalized Fourier transformation of the power
sent in CPW1 Pe(k).
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Rousseau et al.
isolate electrically the CFB stripe from the CPWs, a 100 nm
thick layer of Al2O3 is deposited by sputtering.
The CPWs are designed to be at an angle of 45 to the ferromagnetic stripes to ensure good efficiency in the excitation
and detection of SWs for every angle between the magnetization and propagation direction, as previously demonstrated.8
The width of the CPW lines on top of the CFB is 400 nm.
Thus, given the 45 geometry, the main corresponding wavelength17 is 2.2 lm as the propagation direction in our experiments is along the length of the stripes. For comparison, some
samples with the classical 90 geometry were also measured
and in that case the width of the CPW lines has been increased
to 540 nm to keep the same wave vector distribution as represented in Figure 1(c). We perform experiments in the field domain using a frequency generator at CPW1 with a frequency fe
and a power Pe. The measurements are done with a spectrum
analyzer at CPW2 to analyze the received power for different
frequencies of detection fd. An in-plane magnetic field l0 H of
up to 170 mT is applied perpendicularly to the CFB stripes
in the so called magnetostatic surface wave (MsSW) configuration. The measurements are performed at room temperature.
In Figure 2(a), when the antennas are at 45 , the generated second harmonic at 13 GHz shows two areas with high
amplitude oscillations. The oscillations at high field correspond to the case where the detection frequency, here 13
GHz, is the fundamental SW frequency. The oscillations at
low field correspond to the case where the excitation frequency, here 6.5 GHz, is the fundamental SW frequency. At
90 in Figure 2(b), only the high field oscillations can be
FIG. 2. The ferromagnetic media is composed of one stripe of width 6 lm.
(a) The red continuous curve corresponds to the second harmonic (13 GHz)
power received vs the applied magnetic field when pumping with a power of
17 dBm at 6.5 GHz. The separation D between antennas is 10 lm. The
antennas are at 45 of the stripe as in Fig. 1(a). (b) Same with the antennas
at 90 of the stripe and a power of 20 dBm.
J. Appl. Phys. 115, 053914 (2014)
observed. Therefore, one can make an assumption that we
have two kinds of second-harmonic generation processes.
The first (at higher field) can be detected with any geometry
for the antennas, the other one (lower field) requires some
geometrical condition for its detection.
We first analyze the case when the applied magnetic
field is around that for the fundamental SW mode at 13 GHz,
i.e., an applied magnetic field around 0.11 T. The results are
shown in Figure 3(a). The signal from direct transmission
fd¼fe presents small oscillations with a huge baseline. The
other one from nonlinear excitations at fd¼2fe has high oscillations with a small baseline. The presence of oscillations is
known to be a sign of transmission. Indeed in case of fixed
frequency measurement, an increasing H decreases the wave
vector k selected and the phase delay kD of the SW received.
Thus we can also calculate a finite group velocity9,17,18
FIG. 3. The ferromagnetic media is composed of one stripe of width 6 lm
with the antennas at 45 of the stripe (a). The red continuous curve corresponds to the second harmonic (13 GHz) power received vs the applied magnetic field when pumping with a frequency of 6.5 GHz and a power Pe of 17
dBm. The dashed curve corresponds to the direct resonant excitation at frequency fe¼ fd ¼ 13 GHz and a power of 0 dBm. The distance between
CPWs is 10 lm. (b) Spin-wave dispersion. The open diamonds correspond
to the measured values. The solid line corresponds to the expected dispersion for a wavelength of 2.2 lm in a thin film with a MsSW configuration.
(c) Group velocity in function of the resonant frequency. The diamonds correspond to the measured values. The solid line corresponds to the expected
repartition for a wavelength of 2.2 lm in a thin film and in MsSW configuration. (d) Variations of the SW power received at fd ¼ 13 GHz for an excitation at fe ¼ 6.5 GHz, magnetic field l0 H¼113 mT, and a power Pe. The line
is the fit to the data, the received power varies as I2f ¼ Pe2.8.
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Rousseau et al.
J. Appl. Phys. 115, 053914 (2014)
during the transmission. The dispersion curve is shown in
Figure 3(b), where both experimental points and ideal dispersion19 inside a thin layer for a wavelength of 2.2 lm are represented and are in good agreement. The dispersion formula
for MsSW is given by
x2 ¼ x0 ðx0 þ xM Þ þ
x2M
ð1 e2kt Þ:
4
(1)
Thus, the group velocity vg at resonance, represented in
Figure 3(c), is determined by
vg ¼
dx x2M t 2kt
¼
e :
dk
4x
(2)
In Eqs. (1) and (2), x0¼ cl0 H, xM¼ cl0MS, k is the wave
vector, c/2p ¼ 27.5 GHz/T, l0MS ¼ 1.5 T, x is the circular
frequency of resonance, and t is the thickness of the ferromagnetic media.
As the 6.5 GHz excitation is not propagating, the second
harmonic wave at 13 GHz has to be generated under CPW1, by
magnon confluence as previously reported.20–22 We note there
that the nonlinear signal have same oscillations period as the
direct one. If one only measures a nonlinear generation, one will
obtain peaks from the dispersion of Figures 3(b) and 1(c). In
fact, the small baseline observed at fd ¼ 2fe is caused by our frequency generator having a small component at 2fe with 70 dB
less power than at fe. Thus, the spectrum analyzer measures a
power I ¼ V2directþ 2VSW(H)* Vdirectþ V2SW(H) where I is the
power measured Vdirect the voltage caused by the direct coupling
at the excitation frequency, VSW(H) the voltage arising from
propagation, which phase varies with H, and generates the
power oscillations. Hence, we can compare the period of the second harmonic oscillations to that obtained with a direct excitation at 13 GHz. The oscillations generated by propagation have
the same period in both cases, thus it can be concluded that both
have the same velocity. As the group velocity, frequency, and
field of resonance are the same, it demonstrates that the fundamental excitation can be generated nonlinearly by an excitation
field at half the frequency. Note that the small contribution of
the component at 13 GHz in the excitation at 6.5 GHz has a negligible contribution and cannot generate a detectable SW by
itself. The results presented in Figure 3(d) show that I2f scales as
Pe2.8. The exponent of 2.8 is slightly different from the value
reported elsewhere13 using Brillouin light scattering (BLS) but
this was for another kind of mode and in a finite structure without any propagation. As the oscillations are quite fast, one can
estimate the amplitude VSW of the received SW. For an excitation of 20 dBm at 6.5 GHz, the amplitude of the SW at 13 GHz
is similar to that obtained for a direct excitation at 0 dBm.
Next, we confirm similar behavior at higher harmonic
excitations. In Figure 4, the ferromagnetic media is composed of four stripes of width 1.5 lm separated by 1.5 lm to
avoid any dynamical coupling between them. The distance
between the CPWs is 10 lm and the ferromagnet is 10 nm
thick. We excite at either 3 GHz at 20 dBm or 9 GHz at 0
dBm, and we are interested in the signal received at 9 GHz.
Here, again the oscillations have the same period for the nonlinear third harmonic and for direct excitation. The ratio of
the oscillation amplitudes over the baseline of the nonlinear
FIG. 4. The green continuous curve corresponds to the power received at
fd ¼ 3fe¼9 GHz. The dashed black curve corresponds to the direct resonant
excitation at the same frequency. The antennas are at 45 to the stripes.
excited fundamental wave is slightly smaller than the unity
but can be increased with a more monochromatic frequency
generator or by increasing the thickness of the CoFeB layer.
The excitation of the fundamental mode by lower frequency
(half or third) using the nonlinear effect occurs in all our
devices independently of geometry (45 or 90 ), the width of
the ferromagnetic stripes (6 lm, 1.5 lm) or their thickness
(10, 20 nm). This confirms that this nonlinear process is
related to magnon confluence.20,22
We now address the signal transmitted to CPW2 when the
applied magnetic field is around that for the fundamental SW
mode at 6.5 GHz. In Figure 5(a), the excitation in CPW1 is a
continuous wave at 6.5 GHz with a power Pe of 17 dBm. The
CFB stripe is 6 lm wide. The signal transmitted has two components: one from direct transmission fd¼ fe where it presents
FIG. 5. The ferromagnetic media is composed of one stripe of width 6 lm. (a)
The distance between CPWs is 10 lm. The red continuous curve corresponds
to the second harmonic (13 GHz) power received vs the applied magnetic field
when pumping with a frequency of 6.5 GHz and a power Pe of 17 dBm. The
open diamonds correspond to the power received at 6.5 GHz at the same time.
The antennas are at 45 to the stripe. (b) Variation of the SW power received
at fd ¼ 13 GHz for a resonant excitation at fe ¼ 6.5 GHz, a magnetic field
l0 H ¼ 22 mT, and a power Pe. The line is the linear fit to the data.
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Rousseau et al.
small oscillations with a huge baseline, the other from nonlinear excitations at fd¼2fe where a higher number of oscillations
is also observed with a small baseline. In Figure 5(b), we analyzed the received power from SW propagation at 13 GHz for
different excitation power. We found the same linear dependency that was reported using BLS measurements in Ref. 12 in
similar conditions. In this article, the authors explain that this
second harmonic wave cannot be generated by magnon confluence and has a nonresonant character. They also provided an
explanation for the origin of this second harmonic. In Figure 6,
we remind the origin of this nonlinear mode as well as the reason why it can be detected with antennas at 45 and not at 90
as shown by the Figure 2.
As the excitation of the second harmonic has a nonresonant character, it should be produced by a double frequency
magnetic field. Such a magnetic field arises from the elliptical
precession of magnetization in the propagating MsSW. By taking into account the conservation of magnetization, one can
express the dynamic magnetization along the y direction of the
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
applied magnetic field: My ðtÞ ¼ MS2 m2x ðtÞ m2z ðtÞ
m2 ðtÞ
m2 ðtÞ
M 1 Mx 2 Mz 2 , where mx(t) ¼ mxcos(xt) and mz(t)
S
S
¼ mzsin(xt) correspond to the dynamic along the length and
the thickness, respectively. In Figures 6(a) and 6(b), we
FIG. 6. (a) Schematic of elliptical precession. Here, magnetization along y
is minimal during precession. (b) Same after a phase shift of p/2. Here, magnetization along y is maximal. (c) Spatial structure of the double frequency
dynamic magnetization of the initial wave along the y axis. The created double frequency magnetic field is represented by the black curves. It has several anti-symmetric planes drawn in white. (d) Profile corresponding to the
generated second harmonic amplitude with an antenna superimposed. This
mode is an anti-symmetric mode along the width and has a wave length 2
times smaller than the initial wave as described by the magnetic field h2x in
(c). Red (blue) corresponds to a positive (negative) dynamic magnetization
along the thickness.
J. Appl. Phys. 115, 053914 (2014)
represented the elliptical precession and the variations at 2x of
my(t) where my(t) is the dynamical part of My(t)
1
ðm2x m2z Þcosð2xtÞ, My0 corresponds to the avermy ðtÞ ¼ 4M
S
age magnetization along y My0 ¼ MS M1S ðm2x þ m2z Þ. In
Figure 6(c), my(t) is represented as well as the dipolar magnetic
field that it generates. The anti-symmetric planes of this dipolar
magnetic field are represented by white lines. This dipolar magnetic field, generated by the MsSW at a frequency of fe, has a
frequency of 2fe, a wavelength two times smaller and an
anti-symmetric plane in the center of the stripe. In the first
order, the resulting dynamic magnetization has the same symmetry. The resulting repartition of the dynamics amplitude at
2x is represented in Figure 6(d). As it can be seen in Figure
6(d), the flux through the antenna is not zero owing to the 45
geometry. However, in case of 90 geometry, the flux will be
the average of dynamics at 2x along the width thus zero as a
consequence of the anti-symmetric plane in the center of the
width. The decrease of the wavelength by a factor 2 is in good
agreement with our measurements. Indeed, the period of the
oscillations at 2fe in Figure 5(a) is half the one for the direct
wave. A change of phase of p/2 of the wave at fd¼ fe generates
a change of p in the phase for the wave at fd¼ 2fe, thus causing
the signal at fd¼ 2fe to have twice the number of oscillations.
The dynamics detected at twice the resonant frequency of excitation exists only because of the initial driving MsSW and can
be detected electrically and characterized when the antennas
are not at 90 of the stripe.
Finally, we discuss the signal ratio coming from SW
propagation over the baseline level. In both experiments, for
direct excitation, the ratio is small (below 0.1) for direct excitation but can become large (greater than 1) for second harmonic excitation. Thus, even when direct excitation can
provide information on transmission, the nonlinear excitation
can provide the same with less spurious coupling between
excitation and detection antenna, after a simple frequency
separation. Indeed, to have a clear signal coming from SW
propagation, one should subtract an important base signal in
case of continuous frequency experiments7,17,18 or time domain study.8,23 Thus, the nonlinear excitations can be useful
to provide knowledge of dynamic magnetization or SW
propagation in various devices.
In conclusion, we investigated the propagation of nonlinearly generated harmonic SWs in microscopic stripes. Two
cases arise. One when the pumping frequency corresponds to
the fundamental SW mode, the propagating fundamental
mode generates a second harmonic excitation that requires
some particular geometric conditions to be detected inductively with CPWs to overcome its zero average amplitude.
This mode is not an eigenwave of the stripe and thus cannot
exist without the pumping SW. The other is when the fundamental resonant frequency corresponds to that of the nonlinear generated harmonic. In that case, the second or third
harmonic corresponds to the fundamental propagating SW.
Nonlinear excitations are interesting as the electrically
detected induced signal at the harmonic frequency depends
mainly on SW propagation and not on a direct coupling with
the excitation part. This behavior is promising for potential
use in logic devices.
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