1 Abstract—With the application of microfabrication
techniques, phononic crystals have been transformed over the past decade: from hand assembled millimeter-to-meter scale crystals consisting of metal balls in water or epoxy, to precisely machined crystals with sub-micron features operating at frequencies in excess of 1 GHz. This paper reviews the contributions of Sandia National Laboratories to micro and nano scale phononic crystal devices including: the integration of piezoelectric transducers, the choice of phononic crystal materials, phononic crystal design, and the application of phononic crystals to radio frequency and thermal management applications.
Index Terms—Acoustic Bandgap, Elastic Bandgap, Microelectromechanical Systems, Microfabrication, Phononic Bandgap, Phononic Crystal, Phononic Crystal Cavity
I. INTRODUCTION hononic crystals are the elastic wave equivalent of photonic crystals, where a periodic lattice of scattering inclusions properly arranged in a host or matrix material
causes certain frequencies to be completely reflected by the structure [1]. This creates bandgaps in the frequency response of a phononic crystal. Over the past decade phononic crystals have been scaled from large hand assembled balls in acoustically lossy materials such as water and epoxy to micro and nano fabricated 2D structures realized in low loss materials operating at frequencies from 10-5,000 MHz [1-12]. Micro/Nano fabrication of phononic crystals has presented many opportunities and challenges. Batch fabrication and the ability to co-integrate piezoelectric transducers with phononic crystals for rapid electrical interrogation have lead to experimentation on a wide variety of geometries and devices [1-6, 9-12]. The ability to suspend 2-D phononic crystals above the substrate using micromachining techniques has allowed for low loss phononic waveguides [1-3] and cavities [4-5] to be demonstrated. The wide range of materials
This work was supported by the Laboratory Directed Research and Development (LDRD) program at Sandia National Laboratories and by the DARPA Chip Scale Spectrum Analyzers (CSSA) Program. Sandia National Laboratories is a multiprogram laboratory operated by the Sandia Corporation, Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.
available [1-12] and the implications of finite membrane thickness [13-14] require careful consideration when designing a phononic crystal for a specific application. Phononic crystals and devices have been fabricated and studied theoretically and experimentally in several material systems including Si-air [5,7,8,9,11], SiC-air[4], Si-W[6] and SiO2-W[1-3]. Phononic band gaps at GHz frequencies have been demonstrated in each of these material systems [8,9,4,6,3] with complete phononic bandgap widths exceeding 10% of the center frequency and band gaps for longitudinal waves in excess of 50%. Solid-Solid phononic crystals realized in Si-W and SiO2-W material systems have demonstrated less sensitivity to lithography when compared to Solid-Air phononic crystals [13], allowing scaling to higher frequency operation important to radio frequency (RF) and thermal applications. Certain devices, such as cavities, requiring very low material damping have been best realized in Solid-Air materials systems using high-Q materials such as Si [5] and SiC [4]. Applications of micro-phononic crystals have been in the areas of acoustic isolation [15], RF cavities and filters [4-5, 15-16], acoustic focusing and imaging [17-19] and in thermal management [7-8]. Phononic crystals operating as acoustic mirrors are applicable to mechanically vibrating structures such as gyroscopes and microresonators [4-5, 15-16]. By strategically placing defects in a phononic crystal, devices such as acoustic waveguides [1-3], cavities [4-5, 15-16] and filters can be formed. For applications in RF communications, phononic crystals can realize cavities with high quality factor, low insertion loss and small size [4-5, 15-16], properties not readily achieved together in existing technologies. Phononic crystals also offer unique methods [1] for coupling phononic cavities together to form higher order filters that can improve filter performance metrics such as insertion loss and shape factor. Phononic crystals have demonstrated acoustic focusing and negative refraction [17-19] which may lead to miniaturization or improved performance in applications such as ultrasound and nondestructive testing. Recently, a nano-fabricated Si-Vacuum phononic crystal with its tailored phonon dispersion [7-8] has demonstrated a >50% thermal conductivity reduction of a Si slab while leaving the electrical conductivity nearly unchanged, potentially leading to improved performance in thermoelectric devices.
Micro and Nano Fabricated Phononic Crystals: Technology and Applications
Roy H. Olsson III1, Maryam Ziaei-Moayyed1, Bongsang Kim1, Charles Reinke1, Mehmet F. Su2, Patrick Hopkins1, Yasser M. Soliman2, Drew F. Goettler2, Zayd C. Leseman2 and Ihab El-Kady1,2
1Sandia National Laboratories, Albuquerque, NM, USA 2University of New Mexico, Albuquerque, NM, USA
P
983978-1-4577-1252-4/11/$26.00 ©2011 IEEE 2011 IEEE International Ultrasonics Symposium Proceedings
10.1109/ULTSYM.2011.0241
Wmiandwiintphintacoph
AP
reswitincarrim
whincintbasma
Thandacochoim con
whcomconfillopacoinc
While much reicro and nano d opportunitiesder bandwidterrogating theononic and phteractions [21]oustic regime ononic bandga
II.
A. Origins of PPhononic bansonance conditth the crystal. clusions with aranged in a sq
mpedance Zm, an
here, E and ρclusion or mterface, a wavesed on the aaterials with a r
his scattering wd/or Mie resonoustic wave apoosing a mat
mpedance mism
The Bragg, nditions are sh
here Vavg is the mposite structnstant, a, and ling fraction oening a phononoustic impedaclusion materi
ecent progress scale phonon
s are emergingdth piezoelece phononic cryhotonic crysta] and for proand the intro
ap materials.
. PHONONIC C
Phononic Bandndgaps are fotions that forbA schematic r
acoustic impedaquare lattice wnd acoustic vel
,, .
ρ are the modmatrix materiae propagating acoustic impereflection coef
will be strongesnance conditionpproaches the ltrix and incl
match is desirab
ΓΧ and ΓΜhown in Fig. 1.
Γ
approximate ature. The pitc
radius of incof the inclusionic bandgap it ance mismatcials as possibl
has been achinic crystals, seg including: thctric transducystals, the abi
als to increaseocessing lightoduction of no
CRYSTAL DESIG
dgaps ormed by Brabid acoustic wrepresentation ance Zi and acowithin a matrilocity Vm is sho⁄
,
dulus and masals. At the within the cry
edance mismatfficient
.
st in the presenns, when the wengths give in usion with a
ble to maximize
Μ, and local
ΓΧ
√
1average acousth of the inclu
clusions, r, deons in the mat
is desirable toch between tle while havin
ieved in realizieveral challenghe need for mucers [20] fility to combie phonon-phott signals in ton-linearity w
GN
agg and/or Mwave propagati
of a 2D array oustic velocityix with acousown in Fig. 1.
(
(
s density of tmatrix-inclusi
ystal will scattch of the tw
(
nce of the Brawavelength of t
(4-6). From (a large acouse scattering.
Mie resonan
(
(
(
(
ic velocity of tusions or latti
efine the volumtrix, r/a. Wh
o have as large the matrix ang low veloc
ing ges uch for ine ton the
with
Mie ion of
y Vi stic
(1)
(2)
the ion tter wo
(3)
agg the 3),
stic
nce
(4)
(5)
(6)
(7)
the ice me hen an
and ity
mismatlarge dthe optibandga
Figure 1
B. So A 2Dmateriacommovelocitiboth thsimultavelocityto the Γmuch crystalsachieviseen inBragg aand solresults bandgasolid (thickneFig. 3 [1-6, 9-greatly solid-aiadvantafirst mliteratucrystalsGHz thcrystal
tch between thdensity mismatimum conditio
ap.
1. Schematic of the Bragg
Solid-Solid vs. AD membrane pals has two dion air-solid pies of the mathe Bragg and aneously, widey matching betΓΧ and ΓΜ Brlower filling
s. This greatlying a given freqn Fig. 2 and Fiand Mie resonalid-air (air-SiOof a plane w
ap-to-midgap r(air-Si) membess [13]. This [1] and the m-12], that solidrelaxed lithog
ir phononic ages that lead t
membrane typure in 2007 s were the firsthreshold in 20devices such a
he materials. Ftch between thon for opening
f a 2D square lattg and Mie resona
Air Solid Phonphononic crystistinct advanta
phononic crysttrix and inclusMie resonanc
ening the phontween the matrragg resonance
fractions thay relaxes the litquency. Both ig. 3, which shance condition
O2) phononic cwave expansioratio of both sbrane phononi
plot confirmsmeasured resultd-solid membragraphy requirecrystals. It to solid-solid p
pe phononic [22]. Likewt membrane ph
008-2009 [3,23as waveguides
From (2), this imhe matrix and
a deep and wi
tice phononic crance conditions.
nonic Crystalstal comprised ages over the tal. First, sion can be wce conditions nonic bandgaprix and inclusioe conditions ovan in air-solithography requof these advan
how the overlans in a solid-solcrystal. Fig.
on (PWE) anasolid-solid (W-ic crystals vs the analysis i
ts reported in tane phononic cements when c
was these phononic crystcrystals repo
wise, solid-solihononic crystal3] and to reali[2,22].
mplies that a inclusions is ide phononic
rystal showing
of two solid much more
because the well matched,
can be used p. The close on also leads verlapping at id phononic uirements for ntages can be apping of the lid (W-SiO2) 4 shows the
alysis of the -Si) and air-. membrane in Fig. 2 and the literature crystals have compared to fundamental
tals being the rted in the id phononic ls to pass the ize phononic
984 2011 IEEE International Ultrasonics Symposium Proceedings
frares
frare
Fiairsd
Figure 2. Plot saction for a 2D Wsonances overlapphononic bandg
lithograph
Figure 3. Plot saction for a 2D asonances cannot
form
igure 4. Plot shor phononic crystensitive to the fi
does the air-solid
showing the BraW-SiO2 phononip at moderate filgaps to be formedhy required in ai
showing the Braair-SiO2 phononit overlap and lara compete Brag
owing the bandgtals. The solid-silling fraction. Od phononic cryst
solid-solid pho
gg and Mie bandc crystal. Note tling fractions, end without the relir-solid phononic
gg and Mie bandc crystal. Note t
rge filling fractiogg phononic band
gap-midgap ratioolid phononic cr
Only at very hightal exceed the peononic crystal.
dgaps vs. filling the Mie and Branabling deep, wilatively difficultc crystals.
dgaps vs. filling the Mie and Bra
ons are required dgap.
o for Si-W and Srystal is much leh filling fractionerformance of the
agg ide t
agg to
Si-ess ns e
The Sandia both PWmeasurembeddThe deCMOSsuspendof Si frSiO2 wmismatmateriaacoustiabove, begin tThe suswide belectric One suspendconstanand Alundercufrequenthe mem
Figure from 2SiO2 mthe pho
III. Pfirst phononicgroup [22] is sWE and finite
red results. ding W inclusevice was fabr compatible prded or released
from under thewere chosen htch (the higheals with a reic velocity miat a r/a ratio
to overlap resuspended memb
bandwidth, slancal characteriza
of the majording the phonnt was 45 micrlN transducersut from the sncy phononic cmbrane from th
5. a) Suspended2007. The phonomatrix. b) Modelononic crystal. T
PHONONIC CRY
c bandgap crysshown in Fig. e difference tiThe phononicsions in a 4 mricated on a Srocesses and md from the sub
e membrane inhere for their est available ineflection coeffismatch, a facof 0.32, the B
ulting in a deepbrane phononicnted piezoelecation using a ne
r challenges wnonic crystal. rons. Thus, ths were over 0side of the scrystals requirehe substrate. W
d membrane miconic crystal is foling (PWE and F
The crystal has a60 MHz.
YSTAL DEVICES
stal device rep5 along with thime domain (c crystal wasmicron thick S
Si substrate usimaterials. Thebstrate by isotrn XeF2 or SF6
high acousticn common seficient of 0.5ctor of 1.3. ABragg and Miep, wide phononc crystal featurctric transduceetwork analyze
with this initia At ~60 MH
he 9 layer phon.5 mm wide,
structure. Thed release holeWhen these rel
cromachined phoormed from W inFDTD) and meaa clearly measure
S ported by the he modeling, FDTD), and
formed by SiO2 matrix. ing all post-e device was ropic etching gas. W and
c impedance emiconductor 8) and low
As predicted e resonances nic bandgap. res integrated ers for rapid er.
al work was z the lattice nonic crystal too wide to
hus, the low es to suspend ease holes
ononic crystal nclusions in a asurements of ed bandgap at
985 2011 IEEE International Ultrasonics Symposium Proceedings
werelcryfoumiincGHbe elim
Figfr
ph FcrygaptheclodiaequpotheW-an ma1.2 Da p
ere too large, lease holes thaystal to be comund both theoicron release hoclusion. Once Hz frequencies
completely uminating the re
gure 6. a) Suspeom 2009. The p
SiO2 matrix. bhononic crystal.
Further investiystal has the op-to-midgap rae velocity mismose to the optimameter and edgual. This leadssible in a sque higher sound-Si phononic c
even higheraintaining a r2 μm.
Devices such aphononic crysta
the bandgap wat were too smmpletely underoretically and ole in the centethe W- SiO2 p [3,25], as sho
undercut fromelease holes all
ended membranephononic crystal b) Modeling (FD The crystal has
igation of Fig. optimum bandatio, for r/a valmatch betweenmal value of 2
ge-to-edge spacds to the minimuare lattice phd velocity in Scrystal shown inr operating frrelatively larg
as cavities andal by removal o
was comprommall did not allorcut. A good c
experimentaler of a 28.8 mihononic crystawn in Fig. 6, t
m the edge of l together.
e micromachinedis formed from
DTD) and measus a measured ban
4 reveals that dgap performanlues near 0.25.
n W and Si is 12. At r/a =0.2cing between tmum lithograp
hononic crystalSi when compan Fig. 7 [6] warequency of ge minimum
d waveguides cor distortion of
mised [24], whow the phononcompromise wlly to be a 7icron diameter als were scaledthe crystals couf the membran
d phononic crystW inclusions in
urements of the ndgap at ~ 1 GH
a W-Si phononnce, in terms This is becau.9, which is ve25, the inclusithe inclusions aphy requiremenl. Coupled wared to SiO2, tas able to achie1.4 GHz whfeature size
can be realizedf the inclusions
hile nic
was 7.5 W
d to uld ne,
tal
n a
z.
nic of
use ery ion are nts
with the eve hile
of
d in s.
Figure from 2
Si matri(FDTDa meas
Fig. 8 crystalpiezoeltransmimateriahave be6 [3]. transmiacoustiwaveguphonon Whiloperatiodimenssolid phquality crystal.[4]. Thfilling fsize anconcepacoustimetal electricin a sm
7. a) Suspended2010. The phonoix in order to ach
D) and measuremsured bandgap fo
shows a wav[1] by removi
lectric couplersission of unitals forming theeen formed in
At GHz ission reachedic energy needuides realized nic crystal inclu
le solid-solid pon and wid
sion when comhononic crystafactor (Q) cav
. Shown in Fighe lattice constfaction is 0.44nd a bandgap
pt of this designic cavity from electrodes and
cally excite themall size. The m
d membrane miconic crystal is fohieve high frequ
ments of the phonor longitudinal w
veguide formeing two rows s. The waveguty at 68 MHze phononic crythe GHz phonfrequencies, h
d a maximumds to be focu
by removingusions at these
phononic cryster bandgaps
mpared to air-soals may offer lvities are to be g. 9 is an air-Stant of the crys
4, resulting in ap centered at ns was first exthe relatively
d piezoelectrice resonator, thumeasured respo
cromachined phoormed from W inuency operation.nonic crystal. Thwaves centered a
ed in a W-SiOof inclusions buide achieves az, owing to tystal. Similar nonic crystal shhowever, the
m of -10 dB, used into the vg only a few high frequenc
tals offer highfor the sam
olid phononic lower loss wheformed from t
SiC phononic cstal is 1.83 mica 200 nm minim
2.25 GHz. xplored in [1], y high acousticc transducers us achieving a onse of the air-
ononic crystal nclusions in a b) Modeling he crystal has at ~ 1.4 GHz.
O2 phononic between two a normalized the low loss r waveguides hown in Fig.
normalized because the very narrow rows of the
cies.
er frequency me limiting crystals, air-en very high the phononic crystal cavity crons and the mum feature The general to isolate an
c loss of the required to very high-Q
-SiC
986 2011 IEEE International Ultrasonics Symposium Proceedings
i
cSf
phachandcav
Figure 8. a) Winclusions from awaveguide show
Figure 9. (a) SEcavity with drivcavity by 5 layerSEM of the SiC/filling fraction of
ononic crystalhieves a Q of 2d has a normalvity from the p
W2 waveguide fa phononic crystwing a normalize
EM of a fabricatve and sense AlNrs of phononic c/air phononic cryf 0.44 (c) SEM
sides w
l cavity is sho2,000 at 2.25 Glized transmisspiezoelectric tra
formed by removtal. b) Measureded transmission
ted silicon carbidN transducers sep
rystals on each systal with a pitchof one of the howalls.
wn in Fig. 10GHz in a very ssion of -10 dB. ansducer and m
ving 2-rows of d response of theof 1 at 68 MHz.
de 10th overtone parated from the side (b) close-uph of 1.83µm andoles with vertica
0. The resonatsmall form fact By isolating t
metal electrode
e
p d al
tor tor the
es,
Figure in Fig. 9
smal a Q facthan a min the results crystal overtonphononmaterialayers quality advantathat maproblemresonatcrystal This isinvestigloss filt
Ovfrom hamateriafrequenwith limin exceenabledacoustitransduof a nutilizinperformare now
The Sanjay
10. Measured re9. The resonatoll form factor and
ctor of 2000 wmore traditionasame fabricatireported in [resonator in
nes in the cavinic crystal lattial is achieved, a much more factor. This p
age of eliminaake utilizationmatic. Experimtor is seen to periods isolat
s not predictgation before ptering.
ver the past decand assembledals, with dimenncies < 1 MHmiting dimensiess of 1 GHz. Md suspension aic loss in a 2ucers for rapid number of latng phononic crmance in resonw being reporte
Aauthors woulRaman and
esponse of the por achieves a Q od has a normaliz
was achieved al overtone SiCion process. T16], where th
ncreases linearity and exponeice constants umaking the adarea efficient
phononic crystating many ofof the resona
mentally, the increase with
ting the cavityted from theophononic crys
IV. CONCLUS
cade, phononicd balls arrangensions greater
Hz, to preciselyions of 200 nmMicromachinin
above the subst2D crystal, incharacterizatio
ttices and marystals for minnators, lenses ed.
ACKNOWLEDG
ld like to thanDr. William
phononic crystal of 2,000 at 2.25 Gzed transmission
in ~10 times C resonator [2This is consiste Q factor of rly with the
entially with thuntil the f.Q prddition of phon
method of intal approach hf the overtone
ator in a filter loss of a phon
h the number y from the tranory and requtals can be ap
SIONS c crystals have
ed in lossy fluithan 1 mm an
y microfabricam and operatingng of phononictrate, removingntegration of pon and experimaterial systemniaturization an
and thermal
GMENT nk Dr. DenniChappell of
cavity shown
GHz in a very n of -10 dB.
smaller size 6] fabricated tent with the f a phononic
number of he number of roduct of the nonic crystal
ncreasing the has the added e resonances or oscillator
nonic crystal of phononic
nsducers [5]. uires further pplied to low
e been scaled id and epoxy nd operating ated crystals g frequencies c crystals has g a source of piezoelectric mental study s. Devices nd improved management
is Polla, Dr. DARPA for
987 2011 IEEE International Ultrasonics Symposium Proceedings
funding under the Chip Scale Spectrum Analyzers Program (CSSA). The authors would like to thank Bob Newgard, Dr. Bob Potter and Chris Conway of Rockwell Collins and Dr. Amy Duwel of Charles Stark Draper Laboratories for discussions on high-Q resonators. The authors would like to acknowledge the staff of the Microelectronics Develop Laboratory at Sandia National Laboratories for fabrication of the phononic crystal devices. Finally, we would like to acknowledge our late friend and colleague Dr. James (Jim) G. Fleming who founded the Sandia phononic crystal effort prior to his passing in 2007.
REFERENCES [1] R. H. Olsson III and I. El-Kady, Meas. Sci. Tech., vol. 20, 012002, 2009.
[2] R. H. Olsson III, I. F. El-Kady, M. F. Su, M. R. Tuck, and J. G. Fleming , “Microfabricated VHF acoustic crystals and waveguides,” Sensors Actuators A , vol.145–146, pp. 87–93, 2008.
[3] R. H. Olsson, M. Su, S. X. Griego, Y. Soliman, D. Goettler, Z. Leseman and I. El-Kady “Ultrahigh frequency (UHF) phononic crystal devices operating in mobile communication bands,” Tech. Dig. IEEE Intl. Ultrasonics Symp., pp.1150-3, 2009.
[4] M. Ziaei-Moayyed, M. F. Su, C. Reinke, I. F. El-Kady, and R. H. Olsson, "Silicon Carbide Phononic Crystal Cavities for Micromechanical Resonators," 2011 IEEE 24th International Conference on Micro Electro Mechanical Systems (MEMS), pp.1377-1381, Jan. 2011.
[5] S. Mohammadi, A. A. Eftekhar, A. Khelif, H. Moubchir, W. D. Hunt, and A. Adibi, “High Q micromechanical resonator in a two dimensional phononic crystal slabs,” Appl. Phys. Lett., vol. 95, 051906, 2009.
[6] Y. M. Soliman, M. F. Su, Z. C. Leseman, C. M. Reinke, I. El-Kady, and R. H. Olsson III, “Phononic Crystals Operating in the Gigahertz Range with Extremely Wide Band Gaps,” Appl. Phys. Lett. 97, 193502 (2010).
[7] P. E. Hopkins, C. M. Reinke, M. F. Su, R. H. Olsson III, E. A. Shaner, Z. C. Leseman, J. R. Serrano, L. M. Phinney, I. El-Kady, “Reduction in the Thermal Conductivity of Single Crystalline Silicon by Phononic Crystal Patterning,” Nano Letters 2011 11 (1), 107-112.
[8] B. Kim, J. Nguyen, C. Reinke, E. Shaner, C. Harris, I. El-Kady, and R. H. Olsson III, “Thermal Conductivity Manipulation in Lithographically Patterned Single Crystal Silicon Phononic Crystal Structures,” Tech. Dig. IEEE Intl. Ultrasonics Symp., 2011, in-press. [9] T. T. Wu, L. C. Wu and Z. G. Huang, “Frequency Band Gap Measurements of Two-dimensional air/silicon Phononic Crystals using Layered Slanted Finger Interdigital Transducers”, J. Appl. Phys., vol. 97, pp. 094916-1-7 (2005).
[10] S. Benchabane, A. Khelif, J.-Y. Rauch, L. Robert, V. Laude, “Evidence for complete surface wave band gap in a piezoelectric phononic crystal,” Phys Rev. E, vol. 73(6), pp. 065601, 2006.
[11] S. Mohammadi, A. A. Eftekhar, A. Khelif, W. D. Hunt, A. Adibi, “Evidence of large high-frequency complete phononic band gaps in silicon phononic crystal plates,” Appl. Phys. Lett., vol. 92, pp. 221905, 2008.
[12] N. Kuo, C. Zuo, G. Piazza, “"Microscale inverse acoustic band gap structure in aluminum nitride," Applied Physics Letters, Vol.95, Issue 9, 2009.
[13] C. M. Reinke, M. F. Su, R. H. Olsson III, and I. El-Kady, “Realization of Optimal Bandgaps in Solid-Solid, Solid-Air, and Hybrid Solid-Air-Solid Phononic Crystal Slabs,” Appl. Phys. Lett. 98, 061912 (2011).
[14] A. Khelif, B. Aoubiza, S. Mohammadi, A. Adibi, V. Laude, “Complete band gaps in two-dimensional phononic crystal slabs,” Phys. Rev. E, vol. 74,046610,2006.
[15] T. T. Wu, W-S. Wang, J-H. Sun, J-C. Hsu and Y-Y. Chen, “Utilization of phononic crystal reflective gratings in a layered surface acoustic wave device,” Appl. Phys. Lett., 94, 101913, 2009.
[16] D. Goettler, M. Su, Z. Leseman, Y. Soliman, Roy H. Olsson, and I. El-Kady, “Realizing f.Q product limit in silicon via compact phononic crystal resonators,” J. App. Phys., in press.
[17] M. Ke , Z. Liu, C. Qiu ,W. Wang ,J. Shi ,W. Wen, and P. Sheng, “Negative-refraction imaging with two-dimensional phononic crystals,” Phys. Rev. B, vol. 72, 064306, 2005.
[18] S. Yang , J. H. Page, Z. Liu , M. L. Cowan, C. T. Chan, and P. Sheng, “Focusing of sound in a 3D phononic crystal,” Phys. Rev. Lett, vol.93, 024301, 2004.
[19] N. Kuo, G. Piazza, “Evidence of Acoustic Wave Focusing in a Microscale 630 MHz Aluminum Nitride Phononic Crystal Waveguide”, IEEE Frequency Control Symposium, pp. 530 - 533, June 2010.
[20] H. Kando, M. Watanabe, S. Kido, T. Iwamoto, K. Ito, N. Hayakawa, K. Araki, I. Hatsuda, T. Takano, Y. Nagao, T. Nakao, T. Toi and Y. Yoshii, “Improvement in temperature characteristics of plate wave resonator using rotated Y-cut LiTaO3 / sin structure,” 2011 IEEE 24th International Conference on Micro Electro Mechanical Systems (MEMS), pp.768-771, Jan. 2011.
[21] M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala and O. Painter, “Optomechanical crystals,” Nature, 462, pp. 78-82, 2009.
[22] R. H. Olsson III, J. G. Fleming, I. F. El-Kady, M. R. Tuck, and F. B. McCormick, “Micromachined Bulk Wave Acoustic Bandgap Devices,” International Conf. on Solid-State Sensors, Actuators, and Microsystems, pp. 317-321, June 2007. [23] R. H. Olsson III, I. El-Kady and M. R. Tuck, “Microscale Phononic Crystals and Devices,” EUROSENSORS 2008, pp. 3-8, Sept. 2008. [24] Y. M. Soliman, M. F. Su, Z. C. Leseman, C. M. Reinke, I. El-Kady, R. H. Olsson III, “Effects of Release Holes on Microscale Solid–Solid Phononic Crystals”, Applied Physics Letters, vol. 97, 081907, 2010. [25] M. F. Su, R. H. Olsson III, Z. C. Leseman and I. El-Kady, “Realization of a Phononic Crystal Operating at Gigahertz Frequencies,” Applied Physics Letters, 96, 05311, (2010). [26] S. Gong, N. Kuo and G. Piazza, “A 1.75 GHz SiC Lateral Overtone Bulk Acoustic-Wave Resonator”, 2011 Solid-State Sensors, Actuators and Microsystems Conference (TRANSDUCERS), pp.922-925, June 2011.
988 2011 IEEE International Ultrasonics Symposium Proceedings