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Design and fabrication of aluminum nitride Lamb wave resonators towards high figure of merit for intermediate frequency filter applications

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A design guideline for one-port aluminum nitride (AlN) Lamb wave resonators (LWRs) working at S0 mode with high performance is reported. A fabricated 252 MHz LWR, with an aperture of 200 μm, 12 fingers, and 1.5 μm thick AlN, is found to have a remarkably high figure of merit (FOM), which exhibits a high ratio of the resistance at parallel resonance (Rp) to the resistance at series resonance (Rs) of 1317 and a corresponding product of the effective coupling coefficient (k2eff) and quality factor (Q) exceeding 52. Consisting of such resonators, a 6-stage ladder filter with a low pass-band insertion loss (IL) of 4.5 dB and steep filter skirts is achieved, offering significant advantage of size savings.
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Design and fabrication of aluminum nitride Lamb wave resonators towards high figure of merit
for intermediate frequency filter applications
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2015 J. Micromech. Microeng. 25 035016
(http://iopscience.iop.org/0960-1317/25/3/035016)
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1. Introduction
Aluminum nitride (AlN) Lamb wave resonators (LWRs) are
promising solutions for future single-chip multi-frequency
recongurable wireless communications. With miniaturized
sizes, low impedance, and perfect compatibility with CMOS,
they have shown great potential to constitute low loss and
channel-select band pass lters, therefore, designing a high-per-
formance LWR is pursued by intensive research efforts [16].
For any resonator in a lter, the performance is assessed
by the gure of merit (FOM), which is the product of the
effective coupling coefcient (k2eff) and the quality factor (Q).
Directly related to k2eff, Qs (Q at series resonance), and Qp
(Q at parallel resonance), the ratio of the resistance at par-
allel resonance (Rp) to the resistance at series resonance (Rs),
i.e., Rp/Rs, can be considered as an alternate FOM for lter
designers which is more intuitional. Filters consisting of reso-
nators with a large Rp/Rs tend to possess low insertion loss
(IL) in the passband, deep notches and steep skirts [7].
Though various techniques have been employed to improve
the performance of LWRs, the signicance of promoting its
Rp/Rs is not fully recognized. For instance, Piazza et al devel-
oped the edge-type AlN LWRs suspended by a pair of tethers
[8, 9]; Lin et al proposed AIN LWRs with biconvex edges to
enhance Q, and they also made contributions to temperature
compensation research [10, 11]; Yantchev et al introduced
grating-type LWRs employing metal as reective gratings
[12, 13]. The ratio of Rp to Rs is typically less than 500, which
deteriorates the passband and the roll-off characteristics of the
lter.
This work is mainly focused on theoretical analysis and
experimental verication of the optimized electrode congu-
ration and the impact of resonator geometry dimensions, with
emphasis on the aperture of interdigital transducers (IDT) and
the number of IDT ngers on the Rp/Rs, demonstrating the
regularity of designing LWRs towards high FOM. Resonators
with various sizes are designed, fabricated, and tested. The
design rules for high Rp/Rs LWRs are identied from the
Journal of Micromechanics and Microengineering
Design and fabrication of aluminum nitride
Lamb wave resonators towards high
gureof merit for intermediate frequency
lter applications
JiLiang, HongxiangZhang, DaihuaZhang, Xuexin Duan, HaoZhang and
WeiPang
State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, 92 Weijin
Road, Nankai District, Tianjin, Prople’s Republic of China
E-mail: weipang@tju.edu.cn
Received 27 August 2014, revised 16 November 2014
Accepted for publication 1 December 2014
Published 11 February 2015
Abstract
A design guideline for one-port aluminum nitride (AlN) Lamb wave resonators (LWRs)
working at S0 mode with high performance is reported. A fabricated 252 MHz LWR, with
an aperture of 200 μm, 12 ngers, and 1.5 μm thick AlN, is found to have a remarkably high
gureof merit (FOM), which exhibits a high ratio of the resistance at parallel resonance (Rp)
to the resistance at series resonance (Rs) of 1317 and a corresponding product of the effective
coupling coefcient (k2eff) and quality factor (Q) exceeding 52. Consisting of such resonators,
a 6-stage ladder lter with a low pass-band insertion loss (IL) of 4.5 dB and steep lter skirts
is achieved, offering signicant advantage of size savings.
Keywords: Lamb wave resonator, gureof merit, intermediate frequency, band pass lter
(Some gures may appear in colour only in the online journal)
J Liang et al
Printed in the UK
035016
JMM
© 2015 IOP Publishing Ltd
2015
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Journal of Micromechanics and Microengineering
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0960-1317/15/035016+10$33.00
doi:10.1088/0960-1317/25/3/035016
J. Micromech. Microeng. 25 (2015) 035016 (10pp)
J Liang et al
2
experimental results. It is proven that the aperture of IDT has
the most prominent effect on Rp/Rs and it is also related to spu-
rious mode excitation. In addition, a moderate number of IDT
are suggested, when lower impedance as well as free spurious
modes are taken into consideration. Finally, a 252 MHz LWR,
with an aperture of 200 μm, 12 IDT ngers, and 1.5 μm thick
AlN, is found to have a remarkable Rp/Rs. A 6-stage ladder
lter consisting of LWRs with such optimized resonators is
achieved, which presents a low IL of 4.5 dB, a high rejection
of 40 dB, and a fast roll-off (180 Ω termination impedances are
applied for each port). Under such an effective guidance for
LWR designing, narrow-bandwidth single-chip multi-band
RF solutions can be envisioned in the near future.
2. Theoretical background
Regarding the displacement of particles, Lamb waves are
classied into symmetric ones (S modes) and anti-symmetric
ones (A modes). Most piezoelectric ceramic transducer (PZT)
systems utilizing elastic waves in plates make use of A0 mode
due to its preferable excitability, while, for AlN resonators,
the lowest order of symmetric mode (S0) are mostly studied,
thanks to its high phase velocity and weak phase velocity dis-
persion [14].
A typical AlN LWR is composed of two indispensable
parts, namely, interdigital transducer (abbreviated as IDT,
which is comprised of periodic metal electrodes) and AlN pie-
zoelectric layer, as illustrated in gure1(a). There are a few
parameters that are dominant, i.e., the thickness of AlN, the
number of IDT ngers, the electrode pitch, and the aperture
of the IDT, which are represented by T, N, p, and L, respec-
tively. When an ac signal is exerted on the electrode, Lamb
waves can be excited due to the inverse piezoelectric effect,
propagating laterally. The S0 mode shape of the resonator
is obtained from simulation by 3D COMSOL nite element
analysis (FEA), as shown in gure 1(b), with p = 20 μm,
L = 200 μm and N = 5. A cross-sectional view (xz-plane at the
exact center of the aperture) is presented by gure1(c), pro-
viding thickness displacement patterns of the S0 mode. Also,
gure1(c) presents the displacement of the X component and
Y component through thickness prole, proving the symmetry
characteristics of S0 mode. (For S modes, the displacement of
the X component is symmetric in relation to the center plane,
while the displacement of the Y component is antisymmetric.)
The resonant frequency of the S0 Lamb mode is determined
by equation(1),
λ
==
f
vv
p2
0
00
(1)
where λ is the wavelength, which is also twice the length of
the electrode pitch, thanks to its periodicity, and v0 is the phase
velocity of the S0 Lamb mode.
2.1. Optimum resonator topology for the maximum k2eff
The effective coupling coefcient results in the efciency
of the conversion between electric energy and mechanical
Figure 1. (a) Schematic illustration of a typical AlN Lamb wave resonator. (b) Top view of the S0 mode shape of the resonator (magnitude
of the displacement components). (c) The cross-sectional view (xz-plane) of the S0 mode and the displacement of X component and Y
component through-thickness prole.
J. Micromech. Microeng. 25 (20 15) 03 5016
J Liang et al
3
energy, which the lter bandwidth and IL are also dependent
on. Electrode congurations have a key effect on k2eff. There
are altogether four kinds of electrode congurations, i.e. only
IDT on the top of the resonator (represented by IDT-open),
IDT on the top but with a oating electrode at the bottom
of the resonator (represented by IDT-oating), IDT on the
top with a grounded electrode at the bottom of the resonator
(represented by IDT-grounded), and IDT on both sides of the
resonator (represented by IDT-IDT) [1517]. To predict the
transduction efciency of the 4 topologies, a 2D COMSOL
FEA is utilized to simulate the electric potentials and the
electric performance of all the four types of electrode con-
gurations, as illustrated in gure2. We use the piezoelectric
physical module and the parameters of the resonators are
summarized in table1. The k2eff for each topology is calcu-
lated as follows
ππ
=
k
f
f
f
f2tan2
s
p
s
p
eff
2
1
(2)
where fs is the series resonant frequency and fp is the parallel
resonant frequency. The IDT-IDT topology offers the max-
imum k2eff, resulting from the strong vertical electric eld and
uniform distribution.
Though it was reported that there might be critical issues
depositing AlN on uneven and dense electrodes [18, 19], these
issues do not exist in low or intermediate frequencies, since
the electrode pitch is large enough that this degradation can
be canceled out. In addition, special etching process can be
applied to ensure the good quality of the deposited AlN which
will be discussed in the next sections. Thus the IDT-IDT
conguration can be employed for intermediate frequency
applications.
2.2. The ratio of Rp to Rs, an alternate FOM
For an LWR, the modied Butterworth Van Dyke (mBVD)
model offers a concise way of analyzing its electrical response,
as shown in gure3. The impedance Z of the resonator varying
with the angular frequency ω can be expressed as
Figure 2. Simulation of the electric potentials and electric performance with (a) IDT-open,(b) IDT-oating, (c) IDT-grounded and (d) IDT-
IDT congurations. The color bar represents the electric potentials, and the arrows represent the vector of electric eld.
J. Micromech. Microeng. 25 (20 15) 03 5016
J Liang et al
4
ω
ω
ω
=+
+++
++++
ωω
ωω
()()
ZR
RRjL
RR jL
() e
jC mjC m
jC mjC m
0
11
0
11
m
m
0
0
(3)
where Re is the resistance induced by electrodes, R0 is the
resistance induced by parasitics [16, 20].
The angular frequency at series resonance (ωs) and parallel
resonance (ωp) and the corresponding quality factors (Qs and
Qp) are respectively dened as follows:
ω
=
LC
1
s
mm
(4.1)
ωω
=+
C
C
1
ps m
0
(4.2)
ω
=
+
Q
RRC
1
()
s
sm
em
(4.3)
ω
=
RRC
p
pm m0(4.4)
The effective coupling coefcient k2eff can be approximately
calculated by
π
kC
C8
m
eff
2
2
0
(5)
At the series resonance, since (1/jωsCm) + jωsLm = 0, and
|1/jωsC0|>> Rm, the resistance at series resonance (Rs) can be
expressed as
π
ω
=+ ×
()
RRR C
kQ
8
1
sem
s
s
2
0
eff
2
1
(6)
At the parallel resonance, since (1/jωpC0) + (1/jωpCm) +
jωpLm = 0, the resistance at parallel resonance (Rp) can be
expressed as
=+
+−
+
ωω
()()
R R
RR
RR
pe
jC mjC
m
0
11
0
pp00
(7)
Since |1/jωpC0|>> Rm and |1/jωpC0|>> R0, equation(7) can
be simplied as
ω
=+
+
()
R R
CRR
1
()
pe
pm
0
2
0
(8)
The value of Re is so small compared to the second term in
equation(8) that the nal form of Rp can be expressed as
ωπω
=
+
××
()
R
CRRC
kQ
1
()
81
p
pmp
p
0
2
0
20
eff
2(9)
Since ωs≈ωp, the ratio of Rp to Rs can be expressed as
⎜⎟
ππ
×× ×
()
R
RkQ
Qk
Q
64 8
p
s
sp
4eff
22
2eff
2
2
(10)
It is clearly shown that the ratio of Rp to Rs is proportional
to the square of the k2eff×Q product, demonstrating that this
parameter can be an indication of FOM. In addition, as the
Q-circle on a Smith chart is dened by Rp and Rs, it can be
considered as a pictorial measurement of the FOM, in other
words, the larger the Q-circle is, the better the FOM will be
[20, 21].
To verify the impact of Rp/Rs on lter performance, the
Advanced Design System (ADS) software is used for simu-
lation. The mBVD equivalent circuit is utilized to represent
the series resonators and the shunt resonators, from which
a 3–3 ladder type lter is built. The geometric dimension of
the resonators follow the one listed in table1 with IDT-IDT
congurations. All the equivalent parameters are calculated as
reported in literature [15] and are summarized in table2. Cm
and C0 are xed, thus k2eff is constant. Lm of shunt resonators
is a little larger than those of series one, making a frequency
shift to form the ladder lter. When Rm is constant, the smaller
theRe (R0), the smaller (larger) the Rs (Rp), and thus the higher
the Qs (Qp). In order to demonstrate the impact of Rs and Rp on
lter performance, ve groups of lters constituted of resona-
tors with various parameters are simulated, and the results are
shown in gure4. In the simulation, the terminal resistance of
each port is 1.5 kΩ.
In lter a, series resonators and shunt resonators both have
small Rs and large Rp. It has a low IL and steep skirts and it is
regarded as the control group. In lter b, shunt resonators have
small Rp, while other parameters are the same with that of
lter a. Analogously, lter c have shunt resonators with large
Rs, lter d have series resonators with large Rs and lter d
have series resonators with small Rp. As shown in gures4(a)
and (c), lters based on resonators with a large ratio of Rp to
Rs have a lower insertion loss, while small Rp of shunt reso-
nators or large Rs of series resonators compromises the IL.
As illustrated by gures4(b) and (d), resonators with a large
ratio of Rp to Rs contributes to steep roll-off in proximity to
the pass band, while large Rs of shunt resonators deteriorates
the roll-off at frequencies lower than the center frequency and
small Rp of series resonators impairs the roll-off at frequencies
higher than the center frequency. Also, lter c and e failed to
offer deep notches due to large Rs of shunt resonators or small
Rp of series resonators.
Table 1. Design parameters of all the 4 types of resonators.
Electrode
conguration
Pitch
(μm)
Aperture
(μm)
Number of
IDT ngers
Electrode
width (μm)
IDT-open 15 100 5 10
IDT-oating 15 100 5 10
IDT-grounded 15 100 5 10
IDT-IDT 15 100 5 10 Figure 3. Schematic of the mBVD model of a LWR.
R
m
L
m
C
m
C
0
R
0
R
e
J. Micromech. Microeng. 25 (20 15) 03 5016
J Liang et al
5
3. Experimental verication and discussion
3.1. Resonator design and fabrication
To test the impact of electrode congurations on k2eff, four
kinds of LWRs are designed and fabricated on the same wafer,
and their parameters are the same as those listed in table1.
Additionally, LWRs with frequencies of 141 MHz, 252 MHz,
and 350 MHz of various geometric dimensions with IDT
patterned on both sides are designed and fabricated, as sum-
marized in table3.
A CMOS-compatible AlN MEMS fabrication process
has been employed, as illustrated by gure 5. It starts by
etching an air cavity directly on the silicon wafer by reac-
tive ion etching, and then it is lled with sacricial layer
by chemical-vapor deposition (CVD). After that, chemical
mechanical polarization (CMP) is used to planarize the sur-
face (gure 5(a)). 200 nm molybdenum is deposited by RF
sputtering and patterned as the bottom electrode by plasma
etching (gure 5(b)). In order to prevent AlN cracks on the
bottom electrode, it is critical to form a slight grade at the
edge of the molybdenum. This can be achieved by adjusting
the etching rate of the photoresist and the molybdenum. After
that, 1.5 μm AlN is deposited on the bottom electrode by RF
sputtering deposition (gure 5(c)). Figure 6 is the scanning
electron microscope (SEM) image of the cross-sectional view
Table 2. Modied Butterworth-Van Dyke equivalent parameters for the simulation of the Rp/Rs impact on lter performance.
Filter No Resonator position Rm(Ω)Re(Ω) R0(Ω) Rs(Ω) Rp(Ω) Rp/Rs
Filter a Series resonators 40 10 2 50 52 963 1059
Shunt resonators 40 10 2 50 52 963 1059
Filter b Series resonators 40 10 2 50 52 963 1059
Shunt resonators 40 10 100 50 16 217 324
Filter c Series resonators 40 10 2 50 52 963 1059
Shunt resonators 40 100 2 140 53 053 379
Filter d Series resonators 40 100 2 140 53 053 379
Shunt resonators 40 10 2 50 52 963 1059
Filter e Series resonators 40 10 100 50 16 217 324
Shunt resonators 40 10 2 50 52 963 1059
Figure 4. Simulation results verifying the impact of Rs and Rp on the performance of ladder lters. (a) Demonstration of the impact of Rp
of shunt resonators on the IL. (b) Demonstration of the importance of Rs of shunt resonators to the shape factor. (c) Demonstration of the
impact of Rs of series resonators on the IL. (d) Demonstration of the importance of Rp of series resonators to the shape factor.
J. Micromech. Microeng. 25 (20 15) 03 5016
J Liang et al
6
of the bottom electrode and the AlN lm. As is indicated, the
deposited AlN lm is uniforml and good c-axis orientation is
guaranteed. Next, 200 nm molybdenum is deposited and pat-
terned as the top electrode, exactly the same as the bottom
electrode (gure 5(d)). Then, AlN is etched by a combination
of Cl2-based plasma etching and potassium hydroxide wet
etching (gure 5(e)). After AlN etch, the bottom electrode can
be accessed. Areas that are etched form the air suspended edge
to reect acoustic waves, and also play a role of via holes for
releasing the sacricial layer. Au is then evaporated and pat-
terned by lift-off, serving as electrical connection and pads.
Finally comes the release of the sacricial layer (gure 5(f)).
3.2. Verication of k2eff and the FOM
The electrical performance of the resonators and lters is
measured in an RF probe station at atmosphere pressure.
Connected to an Agilent E5061B network analyzer, ground-
signal (GS) probes are used to test the resonators. After an
open-short-load calibration, S11 parameters are extracted.
Figure 5. Illustration of fabrication process. (a) Air cavity is etched on silicon wafer and then lled with sacricial layer, after which
CMP is used to planarize the surface. (b) Mo is deposited and patterned as the bottom electrodes. (c) AlN is deposited as piezoelectric
layer. (d)Mo is deposited and patterned as the top electrodes. (e) AlN is etched, forming suspended edges and a release window. (f) Au is
deposited and patterned, which is followed by release.
Figure 6. Cross-sectional SEM micrographs of the Mo and AlN lms.
Table 3. Geometric dimensions of 141 MHz, 252 MHz and 350 MHz resonators.
pitch(μm) aperture(μm)
number of
IDT ngers pitch(μm) aperture(μm)
number of
IDT ngers pitch(μm) aperture(μm)
number of
IDT ngers
15 100 5 20 100 6 35 100 6
15 150 5 20 150 6 35 150 6
15 200 5 20 200 6 35 200 6
15 250 5 20 250 6 35 250 6
15 300 5 20 300 6 35 300 6
15 150 7 20 200 5 35 150 2
15 150 11 20 200 8 35 150 6
15 150 15 20 200 11 35 150 10
15 150 19 20 200 14 35 150 14
J. Micromech. Microeng. 25 (20 15) 03 5016
J Liang et al
7
For the measurement of lters, ground-signal-ground (GSG)
probes are used and an open-short-load-through calibration is
executed before S parameters are obtained.
Resonators with all the 4 types of electrode congurations
are fabricated in the same die. Resonators with IDT-open
congurations show no signal at all, thus their results will
not be shown. For the other three topologies, their electrical
responses are shown in gure7, and their performance is
summarized in table 4. For resonators with only IDT on
top, the impedance is inversely proportional to the resonant
frequency and the gap between the electrodes. As a result,
the impedance is so large that signals cannot be detected,
demonstrating that IDT-open congurations are not suit-
able for intermediate frequency applications. For the rest,
the IDT-IDT congurations are proved to possess the highest
k2eff and the largest ratio of Rp to Rs as well, which is in
good agreement with the prediction by FEA simulation, thus
demonstrating IDT-IDT topology is the best choice for inter-
mediate frequency LWRs.
Experimental results of the Rp/Rs as a function of the aper-
ture (L) and numbers of IDT (N) are concluded in gure8.
The ratio of aperture to pitch (L/p) is of the rst importance to
the Rp/Rs. As shown in gure8(a), for all the three frequency
ranges, the Rp/Rs differs along with L and peaks at the vicinity
of L/p = 10. It is also illustrated by gure8(b) that there exists
an optimized value of N for a high Rp/Rs design. The Rp/Rs
maximizes when the product of N and p approaches 240.
As can be seen in gure1(b), since only a portion of the
IDT work, narrower aperture functions less effectively. Hence
LWRs with a larger L tend to possess a higher Rp/Rs. However,
with an increased L, a spurious mode gets close to the target
mode and becomes stronger, thus degrading the main mode as
well as the Rp/Rs. A 252 MHz resonator (p = 20 μm) with L =
300 μm is simulated, and it presents a strong spurious mode
adjacent to the main mode, in concert with the experimental
results, as illustrated in gure9. In a resonant cavity, the dis-
placement function of travelling wave at resonant frequency
along x-axis is presented by
=
()
uu ik
xi
kyexp( )exp
xx
y0(11.1)
π
λ
π
λ
==
k k
2
,
2
x
x
y
y(11.2)
λ λ= = =
p
n
L
m
mn
2
,
2
,, 1, 3, 5, 7
xy (11.3)
ω
+=k k v
xy
22
0
2
(11.4)
where u0 is the maximum amplitude of the vibration, kx (ky)
is the wave number of the x- axis (y-axis), λx (λy) is the wave
length along the x-axis (y-axis), and ω is angular frequency of
the resonant mode. The main resonant mode emerges when m
= n = 1, while higher order modes come out when m = 3, 5,
7… (n = 1). For a constant p, λx and kx are also constant. When
L increases, ky decreases, and thus the resonant frequency of
higher order modes (ω) decreases. This is why a resonator
with a larger L has a spurious mode closer to the main mode.
These higher modes can be suppressed via apodization tech-
niques; however, these techniques compromise k2eff, failing to
boost the FOM [22]. Methods without harming the Rp/Rs are
still being developed.
Due to N being small, the end effects at the edge of IDT
perturb the distribution of electric eld, thus deteriorating the
functionality of LWR [23]. Conversely, increasing N impairs
the end effects, resulting in a promotion of the capability as
well as a decrease in the impedance. As a result, a larger N is
preferred. However, with N increased, the two nodal points
located at the end of the two tethers are not constraining
enough. As a result, some unwanted resonance modes cannot
be avoided. As demonstrated by gure10, compared to a small
conguration, more spurious modes come out with a bulkier
device [24]. Therefore, a moderate N is suggested, making the
total width just less than 240 μm.
With the above mentioned regularities, a 252 MHz LWR
with a 200 μm aperture, 12 IDT ngers, and 1.5 μm thick
AlN is designed, which demonstrates an Rp/Rs value as high
as 1317 and a corresponding product of k2eff×Q exceeding
52. Figure 11(a) presents the measured frequency charac-
teristics for the resonator and concludes the experimental
results. Figure11(b) is its SEM image. An mBVD model is
also developed to t the experimental response. Thanks to the
high-yield AlN MEMS fabrication platform, a relatively low
Rm is achieved, thus a high ratio of Rp to Rs is obtained.
Figure 7. Measured electrical response of resonators with (a) IDT-oating, (b) IDT-grounded, and (c) IDT-IDT congurations.
Table 4. A summary of the performance of resonators with various
electrode congurations.
Electrode
conguration k2eff (%) Rs(Ω) Rp(Ω)
Ratio of
Rp to Rs
IDT-open No signal
IDT-oating 0.89 624 30388 49
IDT-grounded 0.57 465 9368 20
IDT-IDT 1.79 37 36224 975
J. Micromech. Microeng. 25 (20 15) 03 5016
J Liang et al
8
To test the high FOM of the LWR, a 6-stage ladder type
band pass lter is constructed based on the LWR mentioned
above. Figure12 shows the electrical response measured with
terminations of 180 Ω. The lter rolls off sharply, with a small
shape factor of only 3.3 at 40 dB. It has an IL of 4.5 dB and a
rejection of 40 dB. The form factor of the device is as small as
Figure 8. Experimental results of the Rp/Rs varying with (a) the ratio of the aperture (L) to the pitch (p) and (b) the number of IDT gures.
The ratio of the Rp/Rs is maximum when L/p comes close to 10 and the total width of the resonator approaches to 240.
Figure 9. Illustration of spurious mode caused by large L, with p = 20 µm, L = 300 µm, N = 5.
Figure 10. Illustration of spurious mode caused by increased N, with both p = 15 µm, L = 150 µm, while N = 8 and 20 respectively.
J. Micromech. Microeng. 25 (20 15) 03 5016
J Liang et al
9
880 μm×1150 μm, which is a great advantage over the com-
mercial surface acoustic wave lters.
4. Conclusions
In summary, this work presents a guideline for LWRs design
towards a signicantly high Rp/Rs. It is veried that the ratio of
the aperture of the resonator (L) to the pitch (p) has a primary
effect on the Rp/Rs and it peaks approximately at L/p = 10.
A moderate N is suggested for a lower impedance as well as
spurious-mode-free operation. A 252 MHz LWR complying
with the above regularities shows an impressively high value
of the Rp/Rs. Future research will be focusing on reducing the
motional resistance (Rm) of the resonator, which indicates a
higher value of Rp/Rs, and exploring the theoretical limit of
Rp/Rs of LWRs.
Acknowledgment
This work was supported by Natural Science Foundation of
China (NSFC No. 61176106).
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