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J.T. Stewart and D.P. Chen

Vectron International, 267 Lowell Road, Hudson, NH 03051

**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)
T_{ij}
are the components of the stress tensor,
s_{ij}
are the components of the infinitesimal strain tensor,
u_{j}
are the components of displacement,
b_{j}
are the components of the body force per unit volume,
c_{ijkl}
are the components of the elastic stiffness tensor, and
v_{ij}
are the thermoelastic constants with
_{ij}
representing the thermal expansion constants for the material.
In equation (3) the quantity
(T - T_{0})
represents the small variation of temperature from the ambient value,
T_{0}.

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,
u_{i}
are defined in the usual way and
t_{i}
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,
N^{q}
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
N^{q}
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
A_{j}^{(n)}
are amplitude ratios,
are the decay constants along the
*x _{2}* direction, and
is the straight crested propagation number along the

where is the approximate wave number along

with 2w denoting the width of a strip. With this transformed variably crested solution, the acoustic field,

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

Where

where the normalization constant, N, is given as

with the mass density of the material.

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 - T

The perturbation integral (22) is evaluated as a sum over N elements in the mesh as

where

where (

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 50X50 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.

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.

[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 Arrays of Reflecting Grooves", J. Appl. Phys., Vol. 62, 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.

Figure 1. Geometry and dimensions for SAW chip mounted into ceramic package.

Figure 2. Typical finite element model for mounting stress analysis.

Figure 3. First order temperature behavior as a function of propagation direction on ST-Cut quartz for various mounting schemes.

Figure 4. Table showing relative ranking of each mount type with respect to mounting stress sensitivity.

Figure 5. First order temperature as a function of cut angle for Y-rotated quartz. All mount types plotted together.

Figure 6. Table showing the deviation in first order temperature compensation from free case taken from plot in Figure 5.