"High-Q micro-mechanical resonators in a two

"High-Q micro-mechanical resonators in a two
APPLIED PHYSICS LETTERS 94, 051906 共2009兲
High-Q micromechanical resonators in a two-dimensional phononic
crystal slab
Saeed Mohammadi, Ali Asghar Eftekhar, William D. Hunt, and Ali Adibia兲
School of Electrical and Computer Engineering, Georgia Institute of Technology,
Atlanta, Georgia 30332, USA
共Received 29 October 2008; accepted 14 January 2009; published online 5 February 2009兲
By creating line defects in the structure of a phononic crystal 共PC兲 made by etching a hexagonal
array of holes in a 15 ␮m thick slab of silicon, high-Q PC resonators are fabricated using a
complimentary-metal-oxide-semiconductor-compatible process. The complete phononic band gap
of the PC structure supports resonant modes with quality factors of more than 6000 at frequencies
as high as 126 MHz. The confinement of acoustic energy is achieved by using only a few PC layers
confining the cavity region. The calculated frequencies of resonance of the structure using finite
element method are in a very good agreement with the experimental data. The performance of these
PC resonator structures makes them excellent candidates for wireless communication and sensing
applications. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3078284兴
Structures with periodic variations in their mechanical
properties named phononic crystals 共PCs兲1,2 have gained
much attention in recent years due to their unique frequency
characteristics which cannot be achieved using conventional
bulk materials. A very important property of PCs is the existence of phononic band gaps 共PBGs兲 that are frequency
bands in which mechanical energy cannot propagate through
the structure. PCs with PBGs can be used to filter, confine, or
guide mechanical energy and hence are useful for a variety
of applications including wireless communications and sensing. Of special interest for such applications are planar PCs
with two-dimensional 共2D兲 periodicity in space that can provide a low-loss platform with flexibility in the lattice type,
size, and location of the periodic inclusions. Therefore, it
was recently proposed to use 2D PC slabs 共also called 2D PC
plates兲 in which the acoustic properties of a solid slab with
limited thickness are periodically modified in its two inplane dimensions.3–6 Exemplarily, such structures can form
by etching an array of holes in an appropriate substrate 关for
example, Si on insulator 共SOI兲兴 and then undercutting the
structure underneath the holes to form a membrane that is
supported on the sides.6 The acoustic waves in PC slabs are
confined within the thickness of the slab as they cannot leak
into the air 共or vacuum兲 on top and bottom of the structure
due to the large acoustic mismatch. Therefore, unlike surface
acoustic wave-based PC structures7,8 that can suffer considerable loss due to coupling of acoustic waves to the
substrate,9,10 PC slabs provide a very low loss platform for
implementing PC functionalities. Due to this important advantage, there has been much recent interest in PC slab
structures.3–6,11–15 Besides observation of high attenuations
in the transmission spectrum of specific acoustic modes in
certain crystalline directions,4 complete PBGs 共CPBGs兲 共for
all types of waves and for all propagation directions兲 were
theoretically predicted for solid3 and vacuum 共or air兲5 inclusions in a solid slab. CPBGs were then evidenced experimentally both at low11 共less than 1 MHz兲 and high6 共more
than 100 MHz兲 frequencies. Guiding of acoustic waves in
a兲
Electronic mail: ali.adibi@ece.gatech.edu.
0003-6951/2009/94共5兲/051906/3/$25.00
PC line defects was also predicted theoretically for either
free-standing or supported PC slabs.12 The possibility of
wave guiding in free-standing PC slabs was also evidenced
by experiments very recently.13,14
Due to their ability to confine and guide acoustic waves
with low loss, PC slabs have become excellent candidates for
the implementation of chip-scale integrated wireless devices,
especially for high frequency applications in which PC feature sizes are small, resulting in compact structures. The
availability of complimentary metal oxide semiconductor
共CMOS兲-compatible fabrication facilities makes SOI substrates a natural choice for such PC slabs. However, to
implement individual functionalities needed for integrated
wireless applications, reliable frequency-selective structures
must be implemented. Such structures are usually formed
using resonators with high quality factors 共or high Qs兲. Thus,
the development of high-Q resonators with high resonance
frequencies in PC slabs is an urgent need for the deployment
of PC-based integrated wireless systems. However, up to
now there have been only very limited theoretical studies15
on the properties and quality of PC slab resonators.
In this letter we report the design, analysis, fabrication,
and characterization of high frequency Si micromechanical
resonators with high Qs using the CPBG of a PC slab structure. We show that mechanical energy can be efficiently confined using only a few periods of such PC slab structure. We
believe that the results presented in this letter can lead to
more efficient approaches for designing the micromechanical
devices used in wireless communications and sensing
systems.
The PC structure used in this letter is made by embedding a honeycomb array of cylindrical holes in a thin Si slab,
as shown in the inset of Fig. 1共a兲. The inset of Fig. 1共a兲
shows a unit cell of the PC structure in which d is the thickness of the PC slab, a is the distance between the centers of
the nearest holes in the structure, and r is the radius of the
holes. In the designed structure, the geometrical parameters
are chosen as d = 15 ␮m, a = 15 ␮m, and r = 6.5 ␮m. The
band structure of the thin PC slab 共calculated using the plane
wave expansion method described in detail in Ref. 5兲 is
shown in Fig. 1共a兲. It is clear that this geometry of PC slab
94, 051906-1
© 2009 American Institute of Physics
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051906-2
Mohammadi et al.
FIG. 1. 共Color online兲 共a兲 The band structure of a hexagonal lattice PC of
circular holes in Si slab with r = 6.4 ␮m, a = 15 ␮m, and d = 15 ␮m. A unit
cell of the structure is shown in the inset. r, a, and d represent the holes
radius, the distance between the two closest holes, and the slab thickness,
respectively. 共b兲 Schematic of the designed PC slab resonator in which a
cavity is made in the PC structure by removing four rows 共one period兲 of
holes from the PC structure.
provides a large CPBG with frequency extent of 115 MHz
⬍ f ⬍ 152 MHz allowing for confining mechanical vibrations in a wide frequency range. The basic PC resonator
presented in this letter is formed by removing a period 共four
rows兲 of PC holes from the PC structure, as demonstrated in
Fig. 1共b兲, forming a PC cavity. This cavity is surrounded by
three periods 共12 rows兲 of holes on each side in the x direction, as shown in Fig. 1共b兲, and is considered very large
compared to the wavelength in the y direction.
We analyzed the PC resonator structure using threedimensional finite element method 共FEM兲. Si is considered
to be fully crystalline with its main symmetry axes in the x,
y, and z directions. Since the structure is large and has translational symmetry in the y direction, periodic boundary conditions are used to limit the simulation to only one period in
this direction. Three periods of PC holes are placed on each
side of the PC cavity to provide enough confinement for the
trapped modes in the CPBG. The structure is terminated by
the stress-free boundary condition in the x and z directions.
Our simulations show that the PC slab resonator supports
two symmetric shear-horizontal, a dilatational, and two flexural plate modes. Due to the ease of excitation using interdigital transducers 共IDTs兲 and good frequency distinction
from other slab modes excited by IDTs 共which facilitate the
fabrication and characterization of the PC resonator兲 we only
consider the two flexural modes in the remaining of this
letter. The profiles of the three components of the displacement vector of these two modes along with their frequencies
of resonance are shown in Figs. 2共a兲 and 2共b兲. The resonance
FIG. 2. 共Color online兲 Mode profiles of the displacement vector components
on the structure boundaries and the frequency of resonance of the 共a兲 first
and 共b兲 the second studied mode in the PC cavity. u1, u2, and u3 denote the
displacement vector components in x, y, and z directions, respectively. The
color bar indicates the amplitude of each field component in an arbitrary
unit.
Appl. Phys. Lett. 94, 051906 共2009兲
FIG. 3. 共Color online兲 共a兲 Schematic of the PC slab resonator structure with
excitation and receiving transducers on its two sides. In this schematic, the
cavity region is surrounded by four rows of holes 共one period of the PC兲 on
each side. 共b兲 Top SEM image of a fabricated PC slab resonator with the
transducer electrodes on each side. The cavity region is surrounded by 12
rows 共three periods兲 of holes on each side.
frequency of the first mode is approximately in the middle of
the CPBG at 126 MHz while the resonance frequency of the
second mode is at 149.5 MHz, which is very close to the
upper limit of the CPBG. Since the resonance frequency of
the first mode has enough frequency separation from the
edges of the CPBG, we expect a better confinement of this
mode 共and thus, a higher Q兲 compared to the second resonant
mode.
To experimentally characterize the designed PC slab
resonator, we fabricated it on a SOI wafer with two transducers on its two sides, as illustrated in Fig. 3共a兲, to measure the
transmission of the flexural plate waves through the PC resonator as a function of frequency. The details of the fabrication procedure of the device are discussed elsewhere6 and
will not be repeated here for brevity. A scanning electron
microscope 共SEM兲 image of the fabricated structure is shown
in Fig. 3共b兲. The size of the device in the y direction is
1.2 mm, which is very large compared to the wavelength of
the studied waves 共⬃50 ␮m兲 satisfying the translational
symmetry condition used in the simulation. The structure
is connected to the SOI substrate at its two ends in the y
direction.
Utilizing the appropriate piezoelectric properties of the
ZnO, flexural acoustic waves are launched toward the PC
resonator structure. The transmission through the PC structures is then measured using the second transducer on the
other side of the PC structure. Similar to the resonant tunneling effect in semiconductors,16 we expect peaks in the transmission of acoustic waves through this structure at the frequencies of resonance of the cavity. The measured
transmission is then normalized to compensate for the frequency response of the transducers using a similar procedure
as discussed in Ref. 6.
The normalized transmission profiles at frequencies
around the resonance frequencies of these two modes are
shown in Figs. 4共a兲 and 4共b兲. As expected, two peaks associated with the two flexural resonant modes of the cavity
appear in the transmission spectrum of the flexural waves
passing through the PC structure. A peak in the transmission
profile is centered at 126.52 MHz, which is in a very good
agreement with the predicted resonance frequency of 126.0
MHz of the first studied mode found using FEM. The Q of
the transmission profile 共and hence the first resonant mode兲
is 6300, resulting in a frequency by quality factor product
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051906-3
Appl. Phys. Lett. 94, 051906 共2009兲
Mohammadi et al.
FIG. 4. 共Color online兲 Normalized transmission through the PC cavity slab
structure for 共a兲 first, 共b兲 second studied mode for the structure with three
periods 共12 rows兲 of PC holes on each side of the cavity, 共c兲 first, and 共d兲
second studied mode for the structure with only two periods 共eight rows兲 of
PC holes on each side of the cavity. The peak frequency 共f兲, quality factor
共Q兲, and insertion loss 共IL兲 are given in each figure.
共FQP兲 of 0.8⫻ 1012 Hz, which is among the highest FQPs
共a common figure of merit for micromechanical resonators兲
reported to date for Si micromechanical resonators operating
at atmospheric pressure.17
The measured frequency of the second peak in the transmission is 149.1 MHz, which is again in excellent agreement
with the theoretical value of 149.5 MHz found using FEM.
The Q of the second resonant mode is measured to be 2128,
which is nearly one third of that of the first mode. This is an
expected result as the resonance frequency of the second
mode is much closer to the edge of the CPBG compared to
the first mode.
To evaluate the effect of the number of PC layers on the
Q of the resonant modes of the PC resonator, we also fabricated the structure with only two periods 共eight layers兲 of
holes on each side of the cavity and measured the transmission throughout the CPBG with the same procedure discussed above. The corresponding normalized transmission
profiles of the two studied modes are shown in Figs. 4共c兲 and
4共d兲. As it can be seen in Fig. 4, the Qs of first and second
modes are dropped to 1270 and 680 共as compared to 6300
and 2128 for the previous structure兲. As expected, reducing
the number of PC layers results in higher coupling of the PC
cavity modes to propagating modes of the Si slab and consequently much lower Qs and lower insertion loss 共IL兲.
The FQP of the PC resonator reported here, although one
of the highest FQPs reported for Si resonators in atmospheric
pressure, is still an order of magnitude less than the material
limited FQP and the highest reported FQP in micromechanical resonators.17 However, based on the results presented we
believe much higher Qs can be obtained by increasing the
number of PC layers around the cavity in both x and y directions. Further, air damping plays an important role in the
cavity loss in this frequency range17 and operation at lower
pressure will also lead to considerably higher Q values. Due
to the potentially support loss-free structure of these PC slab
resonators, we expect the material loss to be the main limit
on the achievable Q. The extension of these PC resonators to
higher frequencies 共multiple gigahertz兲 is also possible by
scaling the PC structure. Such high-Q, high-frequency resonators are greatly desired for wireless communication and
sensing applications. Also, simultaneous achievement of
photonic and phononic band gap in the same PC structure
can lead to interesting optomechanical interactions such as
enhanced stimulated Brillouin scattering18 and radiation
pressure excitation of the resonators.19
In summary, we reported here PC slab resonators with
high resonance frequencies 共⬃150 MHz兲 with very high Qs
共up to 6300兲 using a CMOS-compatible fabrication process
in Si. The resulted frequency by quality factor product is
among the highest values reported to date for Si-based PC
resonators operating at atmospheric pressure. We studied the
effect of different number of PC layers on the Q of the PC
structure. The results presented here suggest that PC slab
resonators have the potential to achieve the highest performance among all possible acoustic cavities to be used in
wireless communication and sensing systems.
This work was supported by the National Science Foundation under Contract No. ECS-0524255 共L. Lunardi兲 and
Office of Naval Research under Contract No. 21066WK 共M.
Spector兲. The authors wish to thank Dr. Abdelkrim Khelif for
useful discussions.
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