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Design and fabrication of aluminum nitride Lamb wave resonators towards high figure of merit

for intermediate frequency filter applications

View the table of contents for this issue, or go to the journal homepage for more

2015 J. Micromech. Microeng. 25 035016

(http://iopscience.iop.org/0960-1317/25/3/035016)

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1 © 2015 IOP Publishing Ltd Printed in the UK

1. Introduction

Aluminum nitride (AlN) Lamb wave resonators (LWRs) are

promising solutions for future single-chip multi-frequency

recongurable 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 [1–6].

For any resonator in a lter, the performance is assessed

by the gure of merit (FOM), which is the product of the

effective coupling coefcient (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 signicance 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 reective 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 verication of the optimized electrode congu-

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 identied from the

Journal of Micromechanics and Microengineering

Design and fabrication of aluminum nitride

Lamb wave resonators towards high

gureof merit for intermediate frequency

lter applications

JiLiang, HongxiangZhang, DaihuaZhang, Xuexin Duan, HaoZhang and

WeiPang

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

gureof 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 coefcient (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 signicant advantage of size savings.

Keywords: Lamb wave resonator, gureof 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

25

J. Micromech. Microeng.

JMM

0960-1317

10.1088/0960-1317/25/3/035016

Papers

3

Journal of Micromechanics and Microengineering

CB

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

classied 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 gure1(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 gure1(c), pro-

viding thickness displacement patterns of the S0 mode. Also,

gure1(c) presents the displacement of the X component and

Y component through thickness prole, 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 coefcient results in the efciency

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 prole.

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 congurations have a key effect on k2eff. There

are altogether four kinds of electrode congurations, 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) [15–17]. To predict the

transduction efciency 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 gure2. We use the piezoelectric

physical module and the parameters of the resonators are

summarized in table1. 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

conguration can be employed for intermediate frequency

applications.

2.2. The ratio of Rp to Rs, an alternate FOM

For an LWR, the modied Butterworth Van Dyke (mBVD)

model offers a concise way of analyzing its electrical response,

as shown in gure3. 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 congurations. 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 dened 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

=

+

Q

RRC

1

()

p

pm m0(4.4)

The effective coupling coefcient 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 simplied 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 dened 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 table1 with IDT-IDT

congurations. All the equivalent parameters are calculated as

reported in literature [15] and are summarized in table2. 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

theRe (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 gure4. 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 gures4(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 gures4(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

conguration

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 verication and discussion

3.1. Resonator design and fabrication

To test the impact of electrode congurations on k2eff, four

kinds of LWRs are designed and fabricated on the same wafer,

and their parameters are the same as those listed in table1.

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 table3.

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 sacricial 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. Modied 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 reect acoustic waves, and also play a role of via holes for

releasing the sacricial layer. Au is then evaporated and pat-

terned by lift-off, serving as electrical connection and pads.

Finally comes the release of the sacricial layer (gure 5(f)).

3.2. Verication 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 sacricial 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 congurations

are fabricated in the same die. Resonators with IDT-open

congurations show no signal at all, thus their results will

not be shown. For the other three topologies, their electrical

responses are shown in gure7, 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 congurations are not suit-

able for intermediate frequency applications. For the rest,

the IDT-IDT congurations 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 gure8.

The ratio of aperture to pitch (L/p) is of the rst importance to

the Rp/Rs. As shown in gure8(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 gure8(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 gure1(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 gure9. 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 gure10, compared to a small

conguration, 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. Figure11(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 congurations.

Table 4. A summary of the performance of resonators with various

electrode congurations.

Electrode

conguration 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. Figure12 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 signicantly high Rp/Rs. It is veried 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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