Theoretical analysis on Love waves in a layered structure with a piezoelectric substrate and multiple elastic layers

Abstract

A method is developed to analyze the existence and behavior of piezoelectric Love waves in a multilayered structure consisting of a piezoelectric substrate and multiple elastic layers which are isotropic, nonpiezoelectric materials. The acoustic waves and electric fields in the substrate and the layers are investigated. A general dispersion equation is derived to describe the existence of Love surface waves with respect to phase velocity as a function of normalized layer thickness. An iteration formula for X_N is introduced to describe the mechanical action between the layers and the substrate at the interface. Another formula for ε _LN , the equivalent permittivity of the wave-guide layers, is produced to describe the electric fields in the layers. The dispersion equation including a mass loading on the surface of the top layer is deduced, and a formula for calculating the mass sensitivity of the phase velocity is presented. We also find the dispersion equation with an electric shorted interface and introduce a formula for calculating the electromechanical coupling coefficient K² . Numerical results illustrate the phase velocity, the mass sensitivity of the phase velocity and the electromechanical coupling coefficient as functions of the normalized layer thickness for the Love waves in a layered structure with a polymethylmethacrylate (PMMA) layer and a sputtered SiO ₂ layer on a 90° rotated ST-cut quartz (ST-quartz) substrate.

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Theoretical analysis on Love waves in a layered structure with a
piezoelectric substrate and multiple elastic layers
Jiansheng Liua?and Shitang He
Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China
?Received 30 November 2009; accepted 7 February 2010; published online 7 April 2010?
A method is developed to analyze the existence and behavior of piezoelectric Love waves in a
multilayered structure consisting of a piezoelectric substrate and multiple elastic layers which are
isotropic, nonpiezoelectric materials. The acoustic waves and electric fields in the substrate and the
layers are investigated. A general dispersion equation is derived to describe the existence of Love
surface waves with respect to phase velocity as a function of normalized layer thickness. An
iteration formula for XNis introduced to describe the mechanical action between the layers and the
substrate at the interface. Another formula for ? ¯LN, the equivalent permittivity of the wave-guide
layers, is produced to describe the electric fields in the layers. The dispersion equation including a
mass loading on the surface of the top layer is deduced, and a formula for calculating the mass
sensitivity of the phase velocity is presented. We also find the dispersion equation with an electric
shorted interface and introduce a formula for calculating the electromechanical coupling coefficient
K2. Numerical results illustrate the phase velocity, the mass sensitivity of the phase velocity and the
electromechanical coupling coefficient as functions of the normalized layer thickness for the Love
waves in a layered structure with a polymethylmethacrylate ?PMMA? layer and a sputtered SiO2
layer on a 90° rotated ST-cut quartz ?ST-quartz? substrate. © 2010 American Institute of Physics.
?doi:10.1063/1.3359660?
I. INTRODUCTION
Recently, a growing number of acoustic wave modes
have been employed for various sensor applications.1–3Be-
cause of the high sensitivity to the mass loading on the sur-
face, surface acoustic wave ?SAW? devices are of great in-
terest for chemical and biochemical sensing applications. In
Rayleigh-SAW devices, the substrate particles execute a ret-
rograde elliptical motion in the plane described by the direc-
tion of propagation and the normal to the surface. Rayleigh-
SAW is highly sensitive to the deposited surface mass but,
due to the out-of-plane component of the displacement, has
significant attenuation if the surface supporting the wave is
exposed to a liquid. This is unfortunate since many applica-
tions of current interest involve the deposition of mass from
the liquid phase. However, piezoelectric shear horizontal
?SH? acoustic waves are suitable for the applications in liq-
uid sensors because the particle displacement in the SH
mode is in the plane of the device surface. Here, the SH
waves refer to the waves that are SH polarized and propagate
at the surface of a piezoelectric material, such as surface
transverse waves,4,5Bleustein–Gulyaev waves,6and Love
mode waves.7–9
Love wave exists only when a finite thickness layer is
deposited on a semi-infinite substrate, and the shear bulk
wave in the layer is slower than that in the substrate. Love
wave devices are suitable for liquid sensors with very high
sensitivities due to their acoustic energy concentration within
the layer and a few wavelengths in depth below the surface
of the substrate. In recent years, many researchers have re-
ported their developments on the theory of or experiments on
Love wave sensors. The dispersion curve, which describes a
Love wave propagating in a layered structure, is an impor-
tant tool for analyzing or designing a Love wave sensor. In
previous reports, researchers presented the dispersion equa-
tions of Love waves in structures with one or two elastic
layers on a substrate without piezoelectricity.7,10As the pi-
ezoelectricity of a substrate increases, the approximation of
neglecting piezoelectricity or assuming it is accounted for by
a stiffening effect in the phase velocity, may be less valid.
Therefore, a general dispersion equation describing Love
waves in a layered structure is needed to analyze a low loss
Love device fabricated on a substrate with large piezoelec-
tricity. In this contribution, we will focus on the analysis on
the layered structure with one or more elastic, isotropic, and
nonpiezoelectric layer?s? on a piezoelectric crystal substrate.
Both the substrate and the layers are assumed no damping in
our analysis. With the piezoelectricity of the substrate, the
dispersion relationship of the Love mode waves becomes
more complicated. Some researchers only gave the numeri-
cal curves,11and other researchers presented the dispersion
equation of the layered structures with certain special-class
crystal substrates.12,13In our previous work,14we reported a
general dispersion equation of Love waves in a layered struc-
ture with a guiding layer on a piezoelectric substrate.
However, in a Love mode sensor, a hybrid structure with
more than one guiding layer is often adopted to achieve good
performance such as high mass sensitivity, low insertion loss,
and good temperature stability.15A dispersion equation for
Love waves in a multilayered structure is much-needed in
the design of such sensors. For this contribution, we will
provide a theoretical model for analyzing the Love waves in
a?Electronic mail: liujs98@hotmail.com.
JOURNAL OF APPLIED PHYSICS 107, 073511 ?2010?
0021-8979/2010/107?7?/073511/8/$30.00© 2010 American Institute of Physics
107, 073511-1

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a multilayered structure. In Sec. II, the expressions of Love
waves and electric fields in the multilayered structure are
listed. In Sec. III, the boundary conditions are introduced to
eliminate the undetermined coefficients in the expressions,
and a dispersion equation is given. The equivalent mechani-
cal action XNand the equivalent permittivity ? ¯LNare defined
to describe the acoustic waves and electric fields in the lay-
ers. In Sec. IV, the dispersion equations for the mass loading
surface case and the electric shorted interface case are intro-
duced. The mass velocity sensitivity and the electromechani-
cal coupling coefficient are defined. The illustrative results of
a structure with a polymethylmethacrylate ?PMMA? layer
and a sputtered SiO2layer on a 90° rotated ST-cut quartz
?ST-quartz? substrate are presented in Sec. V. The results
obtained in this paper will enrich the theory analysis and
benefit the optimization of Love mode sensors.
II. THE STATEMENT OF THE PROBLEM
The layered structure considered in this paper consists of
N elastic layers deposited on a piezoelectric substrate ?Fig.
1?. A rectangular Cartesian coordinate system ?x1,x2,x3? is
chosen in such way that the x1axis is in the direction of Love
wave propagation, the x2axis is parallel to the direction of
particle polarization, and the x3axis is normal to the surface
of the substrate. The substrate is a piezoelectric crystal filling
in the half-space x3?0. The elastic wave-guide layers, which
are all assumed to be isotropic materials in this paper, are
rigidly linked to the substrate and to each other. The nth
layer has a thickness hn, a mass density ?Ln, a transverse
module ?Lnand a permittivity ?Ln. The nth layer occupies
the space Hn+1?x3?Hn, where Hn=?j=n
from the substrate surface to the interface between the nth
layer and the ?n−1?th layer. The domain above the top layer
is air or vacuum without any mechanical contact with the
layer.
In Love mode sensors, Love wave devices are often fab-
ricated on substrates in which piezoelectric SH waves exist.
For the most general piezoelectric materials, the electric field
is coupled with the three components of the particle displace-
ment vector. However, in this paper we are especially inter-
ested in the piezoelectric crystal whose electric field is
coupled with the particle motion in the x2direction while
uncoupled with those in the x1and x3directions. In general,
the piezoelectric SH-mode waves can propagate along a cer-
tain direction with appropriate cuts of crystals in some crys-
tal classes.16It is interesting to know the conditions of the
material parameters such that only the shear particle motion
Nhjis the distance
is coupled with the electric field. Much research has been
carried out, and the SH wave existence condition is that the
elastic and the elastic matrix c and piezoelectric matrix e of
such a material must obey the following structures:13
c =?
e =?
0
e23
0
e34
0
In such a piezoelectric material, the particle motion is
governed by the Newton’s law; no external charge appears
inside, and the quasistatic approximation is adopted. We as-
sume no energy loss while the acoustic waves propagate
along the positive direction of the x1axis. It is well known
that the particle motion and the electric field in such a piezo-
electric medium satisfy the following equations:
c11 c12 c13
c12 c22 c23 c24 c25 c26
c13 c23 c33
0
c24
0
c15 c25 c34
0
c26
0
0
c15
0
0
c35
0
c55
0
0
c44
0
c46
c46
0
c66?
e36?.
,
0
e12
0
e14
0
e16
e12 e22 e23 e24 e25 e26
?1?
cjk2l
?2u2
?xk? xl
+ ek2l
?2?
?xk? xl
= ??2u2
?t2,
j = 1,2,3;
k,l
= 1,3,
?2?
ek2l
?2u2
?xk? xl
− ?kl
?2?
?xk? xl
= 0,
k,l = 1,3,
?3?
where ? is the mass density and ?klis a matrix of permittivity
constants of the material; u2is the value of the particle dis-
placement in the x2direction; and ? is the value of electric
potential. Equations ?2? and ?3? describe the propagation of
the acoustic wave and its associated electric field inside a
piezoelectric material. They have different solutions in the
substrate, the layers, and the air or vacuum because of their
different material parameters.
A. In the piezoelectric substrate
In this paper, the substrate is a piezoelectric medium
whose electric field is only coupled with particle motion in
the x2direction. The displacement components and electric
potential can be written in the following form:
?
u2= U exp?k?x3?exp?i??t − kx1??
? = ? exp?k?x3?exp?i??t − kx1??.?
?4?
Here U and ? are the magnitudes of the particle motion and
the electric potential; ? is a deep attenuation factor; ? is the
angular frequency; k=2?/? is the wave number; ? is the
wavelength; and i=?−1 is the imaginary unit. In the method
developed in this paper, both the substrate and the layers are
assumed no damping, so k and ? are positive real numbers.
Substitution of the general solution ?Eq. ?4?? into Eqs.
?2? and ?3? produces a homogeneous algebraic set
FIG. 1. The layered structure with a piezoelectric substrate and N elastic
layers.
073511-2J. Liu and S. He J. Appl. Phys. 107, 073511 ?2010?

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?c44?2− 2ic46? − c66+ ?v2
e34?2− i?e14+ e36?? − e16
??U
where v=?/k is the phase velocity of the acoustic waves.
The determinant of the coefficients matrix of the set ?Eq. ?5??
e34?2− i?e14+ e36?? − e16
− ??33?2− 2i?13? − ?11??
??= 0,
?5?
must vanish for nontrivial solutions; thus, a quartic equation
with ? is obtained. When x3→−?, the particle displacement
u2and the electric potential ? must be zero, which means
only the two roots with positive real parts are reasonable
roots. The particular solutions of the acoustic wave and the
electric field in the substrate are consequently in the follow-
ing forms:
?
u2= ?M1A1exp?k?1x3? + M2A2exp?k?2x3??exp?i??t − kx1??
? = ?M1exp?k?1x3? + M2exp?k?2x3??exp?i??t − kx1??
,
?6?
where A1and A2are the ratios of the magnitudes of the
particle displacement to that of the electric potential:
?
A1=?U
A2=?U
??
??
?1
=
?33?1
e34?1
2− 2i?13?1− ?11
2− i?e14+ e36??1− e16
2− 2i?13?2− ?11
e34?2
?2
=
?33?2
2− i?e14+ e36??2− e16
.?
?7?
M1and M2in Eq. ?6? are the coefficients to be determined by
the boundary conditions.
B. In the nth elastic layers
In our analysis, the elastic layers are all assumed to be
isotropic materials without piezoelectricity, which means that
the components containing the piezoelectric element ek2ldis-
appear in Eqs. ?2? and ?3?. The acoustic waves are uncoupled
with the electric field within the layers. The layer thickness is
finite in the x3direction, and the solution for the acoustic
waves in the layer is the following:
u2
Ln= ?M1
Lnexp?ik?Lnx3? + M2
− kx1??,
Hn+1? x3? Hn,
Lnexp?− ik?Lnx3??exp?i??t
?8?
where u2
the transverse propagation constant in the nth layer, and v is
the phase velocity. VLn=??Ln/?Lnis the velocity of the shear
bulk acoustic wave in the nth layer material; ?Lnand ?Lnare
the shear modulus and mass density of the nth layer medium,
respectively.
The electric field in the nth layer can be expressed by
Lnis the particle displacement, ?Ln=?v2/VLn
2−1 is
?Ln= ?M3
Lnexp?kx3? + M4
− kx1??,
Lnexp?− kx3??exp?i??t
Hn+1? x3? Hn,
?9?
where ?Lnis the value of the electric potential. M1
Eqs. ?8? and ?9? are the coefficients to be determined by the
boundary conditions.
Ln–M4
Lnin
C. In the air or vacuum
Only the electric field exists in the air or vacuum above
the layers and the electric potential is zero when x3→+?.
Therefore, the solution of electric field above the layer is in
the following form:
?0= M0exp?− k?x3− H1??exp?i??t − kx1??,
x3? H1,
?10?
where ?0is the electric potential of the electric field distrib-
uted in the air or vacuum. The undetermined coefficient M0
is the value of the electric potential at the surface of the top
layer. H1is the sum of the thicknesses of all layers.
III. THE DISPERSION EQUATION OF LOVE WAVES
In Sec. II, the solutions of acoustic waves and electric
fields are listed out. We will confirm the values of the unde-
termined coefficients in the solutions by the boundary con-
ditions at the interface and the surface of the multilayered
structure. A dispersion equation of piezoelectric Love waves
in such a structure will be established after eliminating the
undetermined coefficients.
A. The mechanical boundary conditions
At the plane where x3=0, the interface of the substrate
and the Nth layer, the particle displacement, and the x3di-
rection stress in the substrate are equal to those in the Nth
layer
M1A1+ M2A2= M1
LN+ M2
LN,
?11?
T1M1+ T2M2= i?LN?LN?M1
T1and T2in Eq. ?12? are the components of the normal stress
acting on the surface particle of the substrate
?
At the plane where x3=Hn=?j=n
nth layer and the ?n−1?th layer, the particle displacement in
the nth layer is equal to that in the ?n−1?th layer:
LN− M2
LN?.
?12?
T1= ?c44?1− ic46?A1+ ?e34?1− ie14?
T2= ?c44?2− ic46?A2+ ?e34?2− ie14?.?
?13?
Nhj, the interface of the
073511-3J. Liu and S. He J. Appl. Phys. 107, 073511 ?2010?

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M1
Lneik?LnHn+ M2
Lne−ik?LnHn= M1
L?n−1?eik?L?n−1?Hn
+ M2
L?n−1?e−ik?L?n−1?Hn,
?14?
and the stress in the x3direction is continuous
?Ln?Ln?M1
Lneik?LnHn− M2
Lne−ik?LnHn? = ?L?n−1??L?n−1?
??M1
L?n−1?eik?L?n−1?Hn− M2
L?n−1?e−ik?L?n−1?Hn?.
?15?
At the plane where x3=H1, none mechanical action con-
tacts the surface of the top layer, thus the normal stress is
zero
M1
Equations ?11?–?16? describe the mechanical boundary
conditions of Love waves in a multilayered structure with
elastic layers on a piezoelectric crystal substrate. By substi-
tuting Eq. ?16? into Eqs. ?14? and ?15? and then analyzing
layer by layer, we can obtain the ratio of the undetermined
coefficients M1
of M1
L1eik?L1H1− M2
L1e−ik?L1H1= 0.
?16?
Lnto M2
LNinto Eqs. ?12? and ?13? produces
Ln. When n=N, substitution of the ratio
LNto M2
−M1
M2
XNin Eq. ?17? is related to the mechanical action be-
tween the layers and the substrate and can be derived from
the following iteration expressions
?
?Ln?Ln− tan?k?Lnhn?Xn−1
=T2− XNA2
T1− XNA1
.
?17?
X1= ?L1?L1tan?k?L1h1?
?Ln?Lntan?k?Lnhn? + Xn−1
Xn= ?Ln?Ln
, n = 2,3, ... ,N.?
?18?
B. The electric boundary conditions
In this section, only the electrically open case is consid-
ered. At the plane where x3=0, the electric potential at the
surface of the substrate is equal to that in the Nth layer, and
the electric displacement in the x3direction is continuous
M1+ M2= M3
LN+ M4
LN,
?19?
D1M1+ D2M2= − ?LN?M3
LN− M4
LN?,
?20?
where ?LNis the permittivity of the Nth layer material. D1
and D2correspond to components of the normal electric dis-
placement of the electric field at the substrate surface
?
At the interface between the nth layer and the ?n−1?th
layer, the electric potentials and the normal electric displace-
ments in the nth layer are equal to those in the ?n−1?th layer:
D1= ?e34?1− ie36?A1− ??33?1− i?31?
D2= ?e34?2− ie36?A2− ??33?2− i?31?.?
?21?
M3
LnekHn+ M4
Lne−kHn= M3
L?n−1?ekHn+ M4
L?n−1?e−kHn,
?22?
− ?Ln?M3
LnekHn− M4
Lne−kHn? = − ?L?n−1??M3
L?n−1?ekHn
− M4
L?n−1?e−kHn?.
?23?
At the surface of the top layer, the electric potential in
the layer is equal to that in air, and the electric displacement
in x3direction is continuous:
M0= M3
L1ekH1+ M4
L1e−kH1,
?24?
− ?0M0= ?L1?M3
?0in Eq. ?25? is the permittivity of air or vacuum.
Equations ?19?–?25? describe the electric boundary con-
ditions of Love waves in the multilayered structure. After a
series of deductions similar to those in Sec. III A, the unde-
termined coefficients M0, M3
we can obtain another ratio of the undetermined coefficients
M1and M2
=D2− ? ¯LN
D1− ? ¯LN
L1ekH1− M4
L1e−kH1?.
?25?
Ln, and M4
Lnare eliminated, and
−M1
M2
,
?26?
where ? ¯LNis the equivalent permittivity of the wave-guide
layers, which can be derived from the following iteration
expressions:
?
? ¯L1= ?L1
?L1tanh?kh1? + ?0
?L1+ ?0tanh?kh1?
?Lntanh?khn? + ? ¯L?n−1?
? ¯Ln= ?Ln
?Ln+ ? ¯L?n−1?tanh?khn?, n = 2,3, ... ,N.?
?27?
Comparing Eqs. ?17? and ?26?, we find the dispersion
equation of Love waves in a multilayered structure:
XN=?D1− ? ¯LN?T2− ?D2− ? ¯LN?T1
?D1− ? ¯LN?A2− ?D2− ? ¯LN?A1
The velocity satisfying the dispersion Eq. ?28? is the
phase velocity of the Love waves propagating in the multi-
layered structure in the case of the electrically open bound-
ary condition. Once the velocity is decided, the undeter-
mined coefficients are determined, and the solution of the
acoustic waves and electric fields in the structure can be
found. In Love wave sensors, viscoelastic mediums are often
adopted as guiding or sensing layers. When a viscoelastic
layer is introduced, the majority of the previous equations
are retained, but the shear stiffness ?Lnof the layer should be
replaced by a complex shear modulus Gf.17The wave num-
ber k in Eq. ?4? should be a complex number, and the veloc-
ity v derived from the dispersion Eq. ?28? is a complex ve-
locity. The existing of the image part of the k and v indicates
the energy attenuation during the propagation of the Love
waves.
.
?28?
IV. MASS SENSITIVITY AND ELECTROMECHANICAL
COUPLING COEFFICIENT
In a Love sensor, the mass sensitivity and insertion loss
are key performance criteria. The mass sensitivity is often
defined as the relative shift in the propagation velocity or the
operating frequency while a mass loading is deposited on the
surface of the guiding layer. Because of the wave-guiding
effect, Love waves are very sensitive to surface perturba-
tions, and a high sensitivity to surface loading can be
073511-4J. Liu and S. He J. Appl. Phys. 107, 073511 ?2010?

Page 5

achieved. The electromechanical coupling coefficient indi-
cates the conversion efficiency from electric energy to me-
chanical energy so it is an important parameter for designing
low loss Love devices. The electromechanical coupling co-
efficient is determined by the relative shift in the phase ve-
locity caused by a circuit-shorted surface of the piezoelectric
substrate. Both the mass sensitivity and electromechanical
coupling coefficient can be derived from the dispersion curve
of a layered structure.
A. The mass velocity sensitivity
The sensitivity to surface mass loading for the acoustic
sensor can be defined as a relative change in phase velocity
due to mass loading of the surface
v??v
?m→0
where ?m is the deposited mass per unit area; v is the phase
velocity without the mass loading perturbation; ?v the ve-
locity shift caused by the mass loading ?m; and Sm
of m2/kg.
The mass loading film must be very thin to allow first-
order perturbation theory to be applied to calculate the
change in the wave velocity. We assume that there is a tiny
mass load deposited on the surface of the top layer and that
the area density of the mass load is ?. Here, the thickness of
the mass load and its influence on the electric field are ne-
glected; thus, the particle motion of the mass load is the
same as that of the surface particle of the top layer. As the
mass loading layer is very thin, the assumption is a good
approach to the case of considering the thickness of the mass
loading layer which was introduced by McHale et al.17,18At
the plane where x3=H1, the particle motion of the mass load
must obey Newton’s law:
L1?x3=H1= −???2u2
x3=H1
Sm
v=1
?m?
,
?29?
vis in units
T3j
L1
?t2?
,
j = 1,2,3,
?30?
where T3j
which acts on the mass load. Substitution of the solution ?Eq.
?8?? into Eq. ?30? produces the following formula:
L1=?L1?u2
L1/?x3is the normal stress of the top layer,
?L1?L1?M1
= − i?kv2?M1
Equation ?31? takes the place of Eq. ?16?; we can obtain
a new dispersion equation that has the same form as Eq. ?28?.
However, the XNhas a new iteration formula instead of ex-
pression ?18?:
?
?Ln?Ln− tan?k?Lnhn?Xn−1
L1eik?L1H1− M2
L1e−ik?L1H1?
L1eik?L1H1+ M2
L1e−ik?L1H1?.
?31?
X1= ?L1?L1
?L1?L1tan?k?L1h1? + k?v2
?L1?L1− k?v2tan?k?L1h1?
?Ln?Lntan?k?Lnhn? + Xn−1
Xn= ?Ln?Ln
, n = 2,3, ... ,N.?
?32?
When the mass load ? is tiny, X1in Eq. ?32? can be
written as:
X1? ?L1?L1tan?k?L1?h1+
?v2
?L1?L1
2??.
?33?
Equation ?33? shows that the velocity shift caused by the
mass load ? equals that caused by the top layer thickness
increasing ?v2/?L1?L1
?v
?m=?v
?h1
?L1?L1
?h1
2. Thus,
?h1
?m=
v2
2
?v
=
v2
?L1?v2− VL1
2?
?v
?h1
. ?34?
In this case, the mass velocity sensitivity can be written
as
Sm
v?1
v??v
?m?
?m→0
=
v
?L1?L1
2
dv
dh1
=
1
?L1
d loge?v2− VL1
dh1
2?
.
?35?
Formula ?35? proves that the sensitivity of a Love sensor
can be enhanced by a lower density of the overlay material.
There exists an optimal layer thickness that results in maxi-
mum sensitivity to surface mass loading. The optimal layer
thickness is determined by the dispersion curve of the lay-
ered structure, which can be gained through numerical cal-
culation or experimental measurement.
B. Electromechanical coupling coefficient
If there is a very thin ideal metal film at the interface of
layer n0and layer ?n0−1?, the electric field will be limited to
the space below the interface. This condition is equivalent to
stating that layer ?n0−1? is an electric conductor. Under this
condition, the permittivity and equivalent permittivity of
layer ?n0−1? tends to infinity, ?L?n0−1?,? ¯L?n0−1?→?. Substitut-
ing ? ¯L?n0−1?into Eq. ?27?, we obtain a new iteration formula
for the wave-guide layer equivalent permittivity
?
If the plane where x3=0, the interface of the substrate
and layer N, is electrically shorted, the permittivity of the
Nth layer tends to infinity ?LN,? ¯LN→?. Substituting this into
?28?, a new dispersion equation in the following form is ob-
tained:
? ¯Ln0=
?Ln0
tanh?khn0?
?Lntanh?khn? + ? ¯L?n−1?
? ¯Ln= ?Ln
?Ln+ ? ¯L?n−1?tanh?khn?, n = n0+ 1,n0+ 2, ... ,N.?
?36?
XN=T2− T1
A2− A1
.
?37?
The electromechanical coupling coefficient is dimen-
sionless and can be determined from the following formula:
K2= 2 ?v − vshort
v
,
?38?
where v, derived from the dispersion Eq. ?28?, represents
the phase velocity of the Love wave propagating in a piezo-
electric multilayered structure. vshort, derived from the disper-
sion Eq. ?37?, denotes the velocity of the Love waves propa-
gating in the same structure but with an electrical short
circuit condition at the plane where x3=0.
073511-5J. Liu and S. HeJ. Appl. Phys. 107, 073511 ?2010?