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Abstract
A general finite element method has
been developed for the analysis of the effects of thermally
induced mounting stresses on the first order temperature
behavior of SAW devices. The present study analyzes this
behavior using a finite element model to obtain an accurate
description of the thermal stresses and deformations in
a crystal plate that has been mounted in various configurations
using an adhesive to bond the quartz into a ceramic
package. This solution is then combined with an analytical
solution of the SAW mode shape for an arbitrary transverse harmonic
in a general perturbation procedure to
obtain the frequency shift. The methods developed herein
are useful for studying the sensitivity of a particular
mounting scheme to thermal stresses for small temperature
excursions. Examples of thermal effects on SAW propagation are
presented for various mounting schemes.
Introduction
When a surface acoustic wave (SAW) device is
mounted in a package, the thermal expansion mismatches
between the SAW substrate, the adhesive material, and the
package cause stresses to be developed in the device
which result in a temperature dependent frequency drift.
This frequency drift is separate from the normal frequency-temperature
behavior which is caused solely by
the temperature dependence of the material constants of
the SAW substrate material. The net effect is a perturbing
of the final packaged device frequency-temperature
behavior, which is easily noticed as a shift of the turn-over
temperature. Experimental evidence exists to suggest that
the exact location of the turn-over temperature for a
mounted device is affected by the bonding agent used, the
package material and geometry, and the location of the
bonding spots in the package. It has also been observed
that the shape of the frequency-temperature curve is also
modified in some cases. The present study seeks to analyze this
behavior using a finite element model to obtain
an accurate description of the stresses and deformations in
a crystal plate that has been mounted in various configurations
using an adhesive to bond the SAW chip into a package. This solution
is then combined with an analytical
representation of the SAW mode shape, along with the
known first order temperature coefficients of the elastic
constants of the substrate material, in a general perturbation
procedure to obtain the frequency shift. A finite element program
has been created for the solution of the
thermal stress problem in the crystal substrate which
allows the properties of the package and bonding agent to
be included. In addition to this, a general analytical solution of
a SAW mode for an arbitrary crystal has been
developed. A general numerical perturbation procedure
has been developed which combines an arbitrary mode
shape with any static finite element solution to obtain the
frequency shift. This integration method is combined with
a sub-meshing algorithm which decomposes an arbitrarily
defined finite element (linear, quadratic, cubic,...) into a
set of linear hexahedral integration cells in which the integral
is evaluated exactly. Similar techniques have been
applied to problems in SAW acceleration sensitivity by
Sinha and Locke [8], as well as by the first author [10,11].
Similar finite element methods have been developed for
BAW devices by Clayton and Eernisse [7] and more
recently by the first author [9] for the analysis of mounting
stresses on crystal resonators.
The developed software is used to analyze the
mounting stress effects on the first order frequency-temperature
behavior for small temperature excursions as a
function of crystal cut and mounting configuration. This
program is used to analyze the effects of thermally
induced mounting stresses in SAW devices, and to compare the various
mounting schemes for thermal stress
related frequency shifts.
Solution of the Biasing State
The development of the static thermoelastic finite
element equations for the solution of the biasing state
begins with the general three dimensional equations of
thermoelasticity
where
and
In equations (1) - (4)
Tij
are the components of the stress tensor,
sij
are the components of the infinitesimal strain tensor,
uj
are the components of displacement,
bj
are the components of the body force per unit volume,
cijkl
are the components of the elastic stiffness tensor, and
vij
are the thermoelastic constants with
ij
representing the thermal expansion constants for the material.
In equation (3) the quantity
(T - T0)
represents the small variation of temperature from the ambient value,
T0.
The variational or weak form of equations (1) - (4)
is formulated for a body occupying a volume
V
bounded by a surface
S
as
where the variational displacements,
ui
are defined in the usual way and
ti
represents prescribed tractions on the
surface of the body. The finite element discretization process
is applied by interpolating the displacements with a
set of shape functions,
Nq
as follows
where
are the nodal displacements. In equation (6) the
superscripts are intended to imply a sum over nodes within
a single element or an entire mesh, depending upon context. This
notation will be employed throughout to save
space. The shape functions
Nq
may take on several forms
and will not be explicitly defined here. The reader is
referred to [15] and [16] for these and other omitted finite
element definitions. Using equation (6) in the functional
(5) gives, in the absence of body forces and applied surface tractions,
Here,
represents the discretized domain with bounding surface
.
For arbitrary variations
,
equation (7) reduces to
where
is the usual stiffness matrix, and
represents the effective nodal force vector. This force vector
represents a set of loads applied to each node in the
mesh to produce the strains which are compatible with the
thermal expansion of the material. The global finite element matrix
system is assembled in the usual way giving
rise to the matrix problem
with solution
Solution of The Mode Shape
The general three dimensional surface acoustic
wave mode shape is obtained from the straight crested
solution obtained by Sinha and Tiersten [3,4,5]
where
C(n)
and
Aj(n)
are amplitude ratios,
are the decay constants along the
x2 direction, and
is the straight crested propagation number along the x1
direction. With this solution, the transformed variably crested
solution is obtained by replacing
with a modified wave number,
, such that
where
is the approximate wave number along x3 for
the mth transverse mode given by
with 2w denoting the width of a strip. With this transformed variably
crested solution, the acoustic field,
uj (x1, x2, x3),
in the transmission path can be written
as a purely real function as
The general solution in the reflector arrays is
obtained by solving the approximate two dimensional surface wave
equations obtained from the variational formulation of Sinha and
Tiersten [5] and using the resulting
transmission matrix in the difference equation solution by
Tiersten, et al. [6], to obtain the effective decay constants
along x1. Using this, the surface wave solution in the
reflector arrays may be written as a purely real function of
the form
Where
1and
1
are decay constants, 
,
,
, and
are constants related to
and
,
as well as scale factors derived from the difference
equation solutions [6]. The variable x'1
is simply a translated x1 for each reflector array.
The normalized mode shape is then obtained as
where the normalization constant, N, is given as
with 
the mass density of the material.
Calculation of The Frequency Shift
The frequency shift under the action of a given
static biasing state is computed using Tiersten's perturbation method
[2] for small fields superposed on a bias [1].
The procedure used here follows closely the methods
employed in [13,14] which considered problems of thermally induced
stresses caused by thick electrodes on bulk
wave devices. In general, the change in resonant frequency,
,
of the 
eigen mode, at frequency
,
is given as
where
and
In equation (22),
are the spatially varying effective elastic constants
derived from the biasing state with
the biasing stresses,
the biasing strains, and
the biasing deformation gradients. The components
represent the small change in the elastic constants at the temperature
T
as defined by equation (26). This definition of
is valid for small temperature
deviations. In general, these constants are a nonlinear
function of
(T - T0),
and for large temperature excursions, the second and third
order temperature derivatives are required, in addition to
the first order term,
contained in equation (26). The values for
used in the present study were obtained
from [12]. At this point it should also be mentioned that
equation (22) is valid only for small temperature excursions. For larger
temperature excursions in the presence of
thermal stresses, higher order elastic constants as well as
the temperature derivatives of the third order elastic constants are
necessary. In equation (23),
are the third order elastic constants for the material and
are the thermoelastic constants defined by equation (4). In
equation (22)
represent the spatial derivatives of
the normalized mode shape given by equation (19).
The perturbation integral (22) is evaluated as a
sum over N elements in the mesh as
where
n,
denotes the volumetric domain of the
nth
element. A single
element integral as defined by equation
(27) is evaluated by first subdividing the element domain,
n,
into an
n1 x n2 x n3
grid of linear hexahedric integration cells and sampling
the spatially varying components of
at the centroid of each cell. For a
sufficiently small cell, the values of
can be assumed to be approximately constant and can thus be
removed from the subintegral. Using this assumption, each
term in the sum of equation (27) can be written as
where
(
1(i),
2(j),
3(k))
are the coordinates of the centroid of the
(ith, jth, kth)
integration cell,
n(i,j,k).
The integral of the normalized mode shape derivatives
appearing on the right hand side
of equation (28) is evaluated exactly using the forms given
by equations (16) and (18). The resulting formulas are
very long and cumbersome and are therefore not listed
here.
Analysis of Results
Figure (1) shows the overall dimensions of a typical SAW chip mounted
into a ceramic package. The
example problem used in the present study consists of a
200 MHz SAW resonator on quartz with the dimensions as
shown in figure (1). The active region of the device consists of a 50
X50
square bounded on each side by a
series of 200 50
wide by /4
long strips, at a spacing of /4.
Figure (2) shows a typical finite element model for
the SAW device mounted into the package (without lid).
The first study considers the effect of thermal stresses on
the first order temperature coefficient of frequency for the
device as fabricated on ST-Cut quartz
(
= 42.75°) as a function of propagation direction on the wafer. Figure (3)
shows the results of this study for 9 different mounting
schemes, labeled 1 - 9. The plots depicted in figure (3)
represent the effective first order temperature coefficient
of frequency (solid curves) compared to the unmounted
behavior (dotted curves). In the analysis that was performed, the
stiffness of the bonding material and ceramic
where increased to exaggerate the differences, hence
approximating a "clamped" case. Therefore the solid
curves represent an upper bound of the effect with the dotted curves
representing a lower bound. Figure (4) shows a
table with a relative ranking of each mounting scheme
from this study. The rank is determined by examining the
deviation at zero degree propagation direction (pure STCut). Figure
(5) shows the results of a study performed on
Y-rotated quartz by varying the
angle from 0 through 90 degrees. In this study, reasonable values were used for the
material properties of the ceramic, lid, and bonding material, taken
from typical industry specifications. The plot
shown reveals a basic 2.5 degree spread in the angle at
which the first order temperature coefficient vanishes. Figure (6)
shows a table with the values of this angle deviation from the free
case for each configuration.
Conclusion
Three dimensional finite element modeling of the
stresses in a SAW device and package assembly allows a
large variety of geometries and mounting schemes to be
studied with the same program, as demonstrated here. The
analysis presented here is most useful for comparing different mounts,
rather than focusing on explicit values of
the frequency shift. Such a tool is very helpful in designing SAW
packaging where stress effects are of concern.
References
[1] J.C. Baumhauer and H.F. Tiersten, "Nonlinear Electroelastic Equations for Small Fields Superposed on a Bias",
J. Acoust. Soc. Am., Vol. 54, No. 4, 1973, pp. 1017-1034.
[2] H.F. Tiersten, "Perturbation Theory For Linear Electroelastic Equations for Small Fields Superposed On a
Bias", J. Acoust. Soc. Am., Vol. 64, No. 3, Sept., 1978 pp.
832-837.
[3] B.K. Sinha and H.F. Tiersten, "Elastic and Piezoelectric Surface Waves Guided By Thin Films", J. Appl.
Phys., Vol. 44 No. 11, Nov 1973, pp 4831-4854.
[4] B.K. Sinha and H.F. Tiersten, "Variational Analysis of
the Reflection of Surface Waves by Arrays of Reflecting
Strips", J. Appl. Phys., Vol. 47 No. 11, July 1976, pp
2824-2832.
[5] B.K. Sinha and H.F. Tiersten, "An Analysis of Transverse Modes in Acoustic Wave Resonators", J. Appl.
Phys., Vol. 51, No. 6, June 1980, pp 3099-3112.
[6] H.F. Tiersten, J.T. Song, and D.V. Shick, "A Continuous Representation of the Acoustic Wave Mode Shape in
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No. 4, August, 1987, pp. 1154-1161.
[7] L.P. Clayton and E.P. Eernisse, "Frequency Shift Calculations for Quartz Resonators",Proceedings of the 1991
IEEE International Frequency Control Symposium, pp.
309-320.
[8] B.K. Sinha and S. Locke, "Acceleration and Vibration
Sensitivity of SAW Devices", IEEE Trans. Ultrason. Ferroelec. Freq. Contr., Vol. UFFC-34, no. 1, pp. 29-38, Jan.
1987.
[9] J.T. Stewart and D.S. Stevens, "Analysis of the Effects
of Mounting Stresses on the Resonant Frequency of Crystal Resonators", Proc. 1997 IEEE International Frequency
Control Symposium, to be published.
[10] J.T. Stewart, R.C. McGowan, J.A. Kosinski, and A.
Ballato, "Semi-Analytical Finite Element Analysis of
Acceleration-Induced Frequency Change In SAW Resonators", Proc. 1995 IEEE International Frequency Control
Symposium, pp. 499-506.
[11] J.T. Stewart, J.A. Kosinski, and A. Ballato, "An Analysis of The Dynamic Behavior and Acceleration Sensitivity of a SAW Resonators Supported By Flexible Beams",
Proc. 1995 IEEE International Frequency Control Symposium, pp. 507-513.
[12] B.K. Sinha and H.F. Tiersten, "First Temperature
Derivatives of the Fundamental Elastic Constants of
Quartz", J. Appl. Phys., 50(4) April 1979, pp. 2732-2739.
[13] H.F. Tiersten and B.K. Sinha, "Temperature Dependence of the Resonant Frequency of Electroded DoublyRotated Quartz Thickness Mode Resonators", J. Appl.
Phys., 50(12) December,1979, pp. 8038-8051.
[14] D.S. Stevens and H.F. Tiersten, "Temperature Dependence of the Resonant Frequency of Electroded Contoured
AT-cut Quartz Crystal Resonators", J. Appl. Phys., 54(4)
April,1983, pp. 1704-1716.
[15] T.J.R. Hughes, The Finite Element Method, Linear
Static and Dynamic Analysis, Prentice Hall, 1987.
[16] O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, Vols. 1&2, McGraw-Hill, 1989.
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