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.
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 , as well as by the first author [10,11]. Similar finite element methods have been developed for BAW devices by Clayton and Eernisse  and more recently by the first author  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
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  and  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
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
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  and using the resulting transmission matrix in the difference equation solution by Tiersten, et al. , 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 . 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  for small fields superposed on a bias . 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
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 . 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 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.
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