Proceeding sof the International Congress on Ultrasonics, Vienna, April 9-13, 2007, Paper ID 1431, Session R07: Bulk and surface
doi:10.3728/ICUltrasonics.2007.Vienna.1431_zhang
acoustic waves
Guided wave spectra and their asymptotic behaviour in multi-layered
plates made of three materials AlN-W-SiO2
Victor Y. Zhang1, Bernard Collet2, Bertrand Dubus1, Michel Destrade2, Tadeusz Gryba1
1
Institut d'Electronique, de Microélectronique, et de Nanotechnologie (IEMN), CNRS, France, victor.zhang@iemn.univ-lille1.fr
2
Institut Jean Le Rond d’Alembert, CNRS/Université Pierre et Marie Curie, Paris, France.
known with more accuracy by selecting the appropriate
interface in the neighbourhood of which the electromechanical energy flux concentrates.
Abstract: A study on guided waves in some layered
plates made of tungsten (W) and SiO2 along with a piezoelectric AlN layer is carried out numerically. Multimode dispersion curves are computed up to sufficiently
high frequency in order to reveal the asymptotic behaviour of guided waves in the short wavelength regime. The
complete spectra were determined without suffering from
numerical instability, by combining the surface
impedance matrix formalism and a recursive algorithm
repeatedly applied at different surfaces/interfaces, and by
preserving the basic matrix size. The results reveal that
wave-confinement and mode-repulsion phenomena appear in certain spectral domains. Their physical origin is
analyzed and interpreted in connection with the characteristic acoustic modes - surface and interface waves at
the boundary of one or two half-spaces of the constituent
materials. Other asymptotic behaviours associated with
classical bulk waves are also highlighted.
The model structures consist of a piezoelectric AlN
layer, a low impedance dielectric SiO2, layer, and a high
impedance metallic tungsten (W) layer. This choice was
motivated by the design of solidly mounted resonators
employing AlN as an active resonator and stacked cells
made of bi-layer SiO2/W as a Bragg mirror to isolate the
resonator from the substrate (Si, for example). The complete acoustic waves spectra are obtained by locating the
zeros of a certain characteristic equation formulated
using either the transfer or the impedance matrix, or the
hybrid of both, as is more convenient. Starting from the
analytical formulation detailed elsewhere [8], we present
here the numerical results obtained for the case where
only the piezoelectric AlN layer and some SiO2/W cells
in the Bragg coupler are kept, in order to facilitate the
analysis of the rather complicated spectrum of guided
acoustic modes.
Key words: guided wave spectra, impedance matrix,
Lamb mode, multilayer plate.
A. Introduction
Engineering applications in surface acoustic wave
(SAW) device design, composite material characterization, and smart structures require an analysis of acoustic
wave interaction with anisotropic and/or piezoelectric
multi layers. More recently, analysis of bulk acoustic
waves (BAW) in multi layered structures composed of
stacked piezoelectric, dielectric and metallic materials
has once again attracted attention of engineers and
researchers working on RF components such as BAW
resonators and filters. The development of efficient
simulation tools is needed to characterize accurately the
electromechanical behaviour of complex stratified structures accounting for realistic electrical and mechanical
interface and boundary conditions.
AlN
(h)
SiO2
(L)
W
(H)
……
Fig.1. Layered plate composed of a piezoelectric AlN layer
together with some additional layers of W and SiO2.
B. Numerical results
We have calculated the solutions for structures consisting of a piezoelectric c-axis oriented AlN layer and a
Bragg coupler having different unit cells of SiO2/W (see
Fig.1). Owing to the isotropy of the coupler materials and
to the special orientation of the AlN layer, the shear horizontal (SH) partial modes are decoupled from the sagittal
plane (SP) ones and are piezo-inactive. In what follows,
we focus only on the SP vibrations which are coupled
with the electrical field. In all numerical examples, the
thickness h of the AlN layer was taken to be 1 Pm. For W
(SiO2), the thickness H (L) was chosen to be Ȝ/4-thick
when the wavelength Ȝ of the longitudinal thicknessmode in AlN is Ȝ=2h; this leads to H=238.79 nm and
L=278.81 nm. The characteristic wave speeds for the
three materials involved in our study are calculated first
and the results are given in Table 1 using some set of
materials constants. They show that AlN is the fastest
Acoustic wave spectra in multi-layers are rather
complex and are difficult to compute, especially in short
wavelength regimes and when the layers have strong
impedance contrast. By repeatedly using the surface
impedance matrix [1]-[7] at different surfaces and/or
interfaces along with a recursive algorithm for the overall
multi layers [3],[4],[6],[7], we were able to determine
numerically the full acoustic spectra for plates made of
up to 4 layers of 3 materials (W, SiO2, and AlN), without
increasing the basic matrix size and up to extremely short
wavelengths [8]. Once the dispersion curves are obtained,
the field distributions of any particular mode can be
-1-
Proceeding sof the International Congress on Ultrasonics, Vienna, April 9-13, 2007, Paper ID 1431, Session R07: Bulk and surface
doi:10.3728/ICUltrasonics.2007.Vienna.1431_zhang
acoustic waves
number of modes is larger (58 instead of 47 in Fig.2)
within the same frequency range (fn d 15). The lowest
limit of the spectra is still the interface mode speed
(though the SAW in massive W is the lowest one among
all possible modes in these materials). Notice that two
branches approach this limit from respectively the upperand lower-side. We interpret this phenomenon by saying
that there exists two interfaces between the W and SiO2
materials within the structure. The lower one is attributed
to the W and the outside SiO2 layers, and the higher one
to the W and the embedded SiO2 layers.
material and W the slowest one.
Fig.2 presents the acoustic spectra calculated for a 3layer AlN/SiO2/W plate. Two branches exist for all frequencies, and higher order branches only appear for f t
fc, with a few of them exhibiting negative slope in the f-k
plot of the dispersion curves near cut-off fc. No mode in
the full spectra is symmetric nor anti-symmetric because
the plate does not exhibit any symmetry in thickness. The
enlarged views of the dispersion curves Vp-f plot, in a
restricted velocity range but in a more extended frequency range (up to fn = 25), allows us to observe more
easily some plateaux where the wave speed seems to
reach an asymptotic limit (horizontal line). An analysis of
the origin of these peculiar behaviours enables us to get a
deeper physical insight into the wave motion in this 3layer plate. A plateau appears clearly in the Vp-f plot for
fn t 3, situated at Vp / V0 | 0.9 which corresponds to the
speed of SAW in massive AlN. Another plateau exists
for fn t 5, situated at the lowest wave speed at Vp / V0 =
0.479 which does not correspond to the speed of SAW in
massive SiO2 nor in W. Here a careful analysis reveals
that the plateau tends to the wave speed of the interfacial,
also called Stoneley mode which would exist and
propagate at the interface of W and SiO2 were both
materials to fill up the half-spaces. Another asymptotic
limit is observed just above the interface mode for fn >
10, which corresponds to the shear BAW in massive W.
At higher frequencies (fn > 17), some branches reach a
plateau-like zone just below Vp / V0 = 0.9 and over a
finite frequency range, which is relative to the longitudinal BAW in massive W; the branches then undergo a
sharp decrease in phase velocity; after that, they approach
first the asymptotic limit of shear BAW speed of massive
SiO2 (Vp / V0 | 0.63) and then the shear BAW speed of
massive W (Vp / V0 | 0.5).
The third lowest mode tends to the SAW speed of
massive SiO2 (Vp / V0 | 0.568) after mode-exchanging
(crossing) near fn|2.2 with Vp / V0 | 0.9, a value near the
characteristic plateau visible for fn>3 and attributed to the
SAW in massive AlN. The plateau relative to the SAW in
SiO2 is well constructed as soon as fn>4. Higher modes
get divided into two families in the very short wavelength
regime. The lower family approaches the shear bulk
speed limit of W (just above the interface mode), and the
upper family tends to the shear bulk speed limit of SiO2
by pair, owing to the couple of SiO2-layer in the
structure. A barely visible single-mode plateau appearing
for fn>15 and just below Vp/V0|0.9 is due to the intermediate confinement originated from the longitudinal bulklike mode in the W-layer. The single mode plateau appearing for fn>7 and just above Vp/V0|1 is due to the
confinement originated from the shear bulk-like mode in
the AlN-layer. The interface mode speed is the bottom
limit of the phase velocity in the whole spectra of guided
waves in the 4-layer AlN/SiO2/W/SiO2 plate. No mode
tends to the SAW speed of massive W in neither 3- and
4-layer cases involving only one W-layer in Figs. 2 and
3, and this is true whether the W is or is not an exterior
layer.
When the roles of the W and SiO2 layers are interchanged in the 4-layer system, the lowest branch is the
SAW speed of massive W (Vp / V0 = 0.454). We point out
that it is not sufficient to have the W-layer as an exterior
one, as in Fig.2, but that another W-layer embedded in
the structure is also required for the SAW in W to exist in
the very short wavelength regime. The next lowest
branch approaches rapidly (for fn t 2) the speed of the
interfacial mode. Another branch also tends to the interface mode speed, but at a much higher frequency, fn t 8.
This phenomenon is logically explained by the presence
of two W/SiO2 interfaces in the current configuration.
The branch which reaches the interfacial wave speed first
is essentially at the interface of SiO2 with the outside Wlayer, while the other one is mainly at the interface of
SiO2 with the embedded W-layer. No mode tends to the
SAW speed of massive SiO2 because this layer has no
stress-free surface.
Another peculiar feature worth noticing is that the
horizontal asymptotic line does not cross any branch
when they approach each other. This is illustrated in
Fig.3 which shows some zoomed views of Fig.2 in some
selected zones. In particular, the line for SAW in AlN
around the very beginning of its existence (corresponding
to the red-circled part) is enlarged in the top-right quarter
of Fig.3. The others two subplots for shear bulk waves in
AlN (at Vp / V0 | 1) and in SiO2 (at Vp / V0 | 0.63) show
clearly that the branches do not cross but instead exhibit a
sharp slop-change whenever they get near one each other.
This feature indicates the energy interchange between
various modes, a property known as mode-repulsion
when many modes co-exist.
If the same three layers are arranged differently, say
as AlN/W/SiO2, a sandwich with the slowest layer embedded between two faster ones, then the plateau attributed to the SAW in massive AlN appears as soon as fn t
2, and a supplementary plateau exists above Vp / V0 = 0.5,
which corresponds to the SAW speed in massive SiO2.
Other characteristics remain rather similar [8].
C. Conclusion
Full acoustic spectra in some plates consisting of
layers having strong impedance and/or velocity contrast
are numerically studied. It is found that in a 3-layer plate
of either AlN/W/SiO2 or AlN/SiO2/W, the lower limit of
the wave spectra is the interface wave existing between
SiO2 and W layers. The asymptotic line standing for the
speed of SAW in an AlN half space always exists, but the
To go further, we also calculated the dispersion
curves, as shown in Fig.4, for a 4-layer plate
AlN/SiO2/W/SiO2. The characteristics of higher-order
modes cut-off, no symmetry, and negative slope are
similar to the previous 3-layer plates, except that here the
-2-
Proceeding sof the International Congress on Ultrasonics, Vienna, April 9-13, 2007, Paper ID 1431, Session R07: Bulk and surface
doi:10.3728/ICUltrasonics.2007.Vienna.1431_zhang
acoustic waves
[2]
line for the SAW in SiO2 only exists in the case of
AlN/W/SiO2. No SAW in W was found whether the Wlayer is or is not an exterior one in the 3-layer plates. In a
4-layer AlN/SiO2/W/SiO2 plate, the 2-fold interface wave
speed remains the bottom limit of the whole spectra,
lying below the SAW speed of massive SiO2, and no
SAW in W was found. When the layers are arranged in
the order AlN/W/SiO2/W, the SAW in W does exist and
becomes the low limit of the whole wave spectra lying
below the 2-fold asymptote associated with the interface
wave between W and SiO2. There is no SAW in massive
W in any structure possessing only one W-layer, even if
it is an exterior layer, and no interface wave exists
between AlN and W or SiO2, though the SAW in AlN
exists anyway. The results made evident in the present
study can be useful for structure optimization in BAW
filter and Lamb wave guide design.
[3]
[4]
[5]
[6]
[7]
D. Literature
[1]
B. Honein, A.M.B. Braga, P. Barbone, G. Herrmann,
"Wave propagation in piezoelectric layered media with
some applications," J. Intell. Mater. Syst. Struct. 2 (1991)
542–557.
[8]
V.Y. Zhang, T. Gryba, J. M. Orellana, B. Collet, "Surface
impedance matrix for the study of acoustical propagation
in multilayered structures," Acta. Acust.- Acust. 88, 218–
230 (2002).
T. Pastureaud, V. Laude, S. Ballandras, "Stable
scattering-matrix method for surface acoustic waves in
piezoelectric multilayers," Appl. Phys. Lett. 80 (2002)
2544–2546.
S.I. Rokhlin and L. Wang, "Stable recursive algorithm for
the elastic propagation in layered anisotropic media:
stiffness matrix method," J. Acoust. Soc. Am. 112 (2002)
822–834.
E.N. Tan, "A concise and efficient scattering formalism
for stable analysis of elastic wave in multilayered
anisotropic solids," Ultrasonics 41 (2003) 229–236.
L. Wang and S.I. Rokhlin, "A Compliance/Stiffness
Matrix Formulation of General Green’s Function and
Effective Permittivity for Piezoelectric Multilayers,"
IEEE Trans. UFFC 51 (2004) 453–463.
B. Collet, "Recursive surface impedance matrix methods
for ultrasonic wave propagation in piezoelectric
multilayers," Ultrasonics 42 (2004) 189–197.
V.Y. Zhang, B. Dubus, B. Collet, M. Destrade,
"Piezoacoustic wave spectra in high impedance-contrast
layered plates, using improved matrix formalisms,"
(submitted for publication).
Table 1 Characteristic wave speeds for the studied materials (SC=short-circuit, OC=open circuit, L, T=longitudinal, shear BAW):
c-axis AlN: SC/OC
X-propagation
W (High-impedance)
SiO2 (Low-impedance)
VL (m/s)
10605/10939
9911 (non-piezo)
5224
6100
Y-polar-VT
5796
5597
2888
3655
2nd VT (m/s)
5796
5796/5867
VSAW (m/s)
5395/5402
5417/5437
2669
3342
Fig.2. Acoustic spectra of guided waves in a 3-layer plate AlN/SiO2/W: global view (left panel) and enlarged partial views of
normalized phase velocity plotted vs. frequency in two particular spectral regions (right panel).
-3-
Proceeding sof the International Congress on Ultrasonics, Vienna, April 9-13, 2007, Paper ID 1431, Session R07: Bulk and surface
doi:10.3728/ICUltrasonics.2007.Vienna.1431_zhang
acoustic waves
Fig.3. Enlarged views of guided waves spectra for the plate AlN/SiO2/W showing the mode-repulsion, namely no branches cross one
each other, near the asymptotes associated to the SAW (bottom-left subplot) and shear BAW in massive AlN (top-right subplot), as
well as the one to the shear BAW in SiO2 (bottom-right subplot). Red ellipses and arrows indicate approximate zoomed regions.
Fig.4. Acoustic spectra of guided waves in a 4-layer plate AlN/SiO2/W/SiO2: global view (left panel) and enlarged partial views of
normalized phase velocity plotted vs. frequency in two particular spectral regions (right panel).
-4-