H. F. Tiersten

An analysis of thickness‐extensional trapped energy resonant device structures with rectangular electrodes in the piezoelectric thin film on silicon configuration

It has recently been shown that highly uniform thin layers can be etched in a small well‐defined region of a silicon wafer and that good quality thin piezoelectric films such as zinc‐oxide can be deposited along with the electrodes to form high frequency trapped energy resonant device structures. The accompanying analytical work has been for pure thickness vibrations only. In this work an analysis of essentially thickness‐extensional trapped energy modes in the thin piezoelectric film on silicon composite structure is performed. It is shown that for small wavenumbers along the plate the dispersion equation is isotropic in the plane of the plate even though the silicon is severely anisotropic in that plane. From the resulting dispersion relation an asymptotic differential equation describing the mode shape along the surface of the composite plate vibrating in the vicinity of a thickness‐extensional resonance is obtained along with the associated edge conditions. Since the mode is essentially thickness‐exte...

A perturbation analysis of the attenuation and dispersion of surface waves

A perturbation formulation of the equations of linear piezoelectricity is obtained using a Green’s function approach. Although the resulting equation for the first perturbation of the eigenvalue strictly holds for real perturbations of real eigenvalues only, it is formally extended to the case of purely imaginary perturbations of real eigenvalues. The extended equation is applied in the calculation of the attenuation of surface waves due to the finite electrical conductivity of thin metal films plated on the surface and air loading. The influence of the viscosity of the air is included in the air‐loading analysis, and the calculated attenuation increases accordingly. Since the metal films are thin compared with a wavelength, an approximate thin‐plate conductivity equation is employed in the determination of the attenuation due to the electrical conductivity of the films. The resulting attenuation is obtained over a very large range of values of sheet conductivity. This is accomplished by using the equatio...

Temperature dependence of the resonant frequency of electroded doubly‐rotated quartz thickness‐mode resonators

A system of approximate equations for the determination of thermal stresses in piezoelectric plates with large thin films of a different material plated on the surfaces is derived. The plate equations are obtained by making a suitable expansion of the pertinent variables in the thickness coordinate, inserting the expansion in the appropriate variational principle and integrating with respect to the thickness in the manner of Mindlin. Conditions resulting in both extensional and flexural stresses are considered and the full anisotropy of the quartz is included in the treatment. The particular case of purely extensional thermal stresses resulting from large electrodes of equal thickness plated on the major surfaces of doubly‐rotated quartz thickness‐mode resonators is treated in detail. The changes in resonant frequency resulting from the thermally induced biasing stresses and strains are determined from an existing perturbation equation. Calculations, using the newly defined first temperature derivatives o...

Elastic and piezoelectric surface waves guided by thin films

A system of approximate two‐dimensional surface‐wave equations and edge conditions in one scalar variable is derived from Hamiltons principle for linear piezoelectric media by assuming suitable depth behavior and integrating with respect to depth. The assumed behavior with depth is determined from the known straight‐crested surface‐wave solutions of the three‐dimensional equations for the plated and unplated substrate in connection with the known variable‐crested solutions for the isotropic substrate. The influence of the inertia, stiffness, and electrical shorting of the film is included in the analysis. The application of the derived equations to the problems for isotropic substrates treated earlier by an ad hoc technique indicates that the derived equations are extremely accurate. Among other things, the analysis reveals that a slot in an aluminum‐oxide film on a T‐40 glass substrate exhibits guiding characteristics twice as good as the combination of aluminum on T‐40 glass in the essentially nondispe...

Second harmonic generation and parametric excitation of surface waves in elastic and piezoelectric solids

Nonlinear electroelastic equations, quadratic in the small field variables, are applied in the analysis of second harmonic generation of surface waves in piezoelectric solids. Preliminary to the treatment of the anisotropic piezoelectric case, the more tractable problems of the second harmonic generation and parametric excitation of surface waves in isotropic elastic solids are treated. In all instances the solutions asymptotically satisfy the nonlinear differential equations and boundary conditions on the surface of the semi‐infinite solid to a specified order in a small parameter. Since the equations are quadratic rather than cubic in the small field variables, only initial spatial rates of growth of the harmonically generated and parametrically excited waves are determined. Nevertheless, an extension of the analysis to enable the calculation of more than the aforementioned initial slopes is indicated. In the isotropic elastic case the solution for the second harmonic reveals, among other things, that t...

Intrinsic stress in thin films deposited on anisotropic substrates and its influence on the natural frequencies of piezoelectric resonators

A description of intrinsic stress in thin films deposited on anisotropic (piezoelectric) substrates is obtained from the rotationally invariant equations of nonlinear elasticity (and electroelasticity). The accompanying residual stress state in the substrate is, of course, included in the description, as is the residual stress state in the thin film. The equations for small dynamic fields superposed on the static (intrinsic plus residual) bias are obtained, and the equation for the perturbation in eigenfrequency of a piezoelectric resonator due to the intrinsic and residual stress state is presented. The static biasing equations for both extension and flexure of the plated crystal plate, which are required for the determination of the residual stress state from the intrinsic stress state, are obtained. The important case of a thin film deposited on one side of a plate substrate is treated in detail, and the relation of the flexural curvatures of the plated plate to the components of the intrinsic stress i...

One‐dimensional acceleration waves and acoustoelectric domains in piezoelectric semiconductors

The theory of acceleration waves is applied in the analysis of the formation and propagation of acoustoelectric domains in piezoelectric semiconductors. A one‐dimensional version of the general nonlinear electroelastic equations for deformable semiconductors recently presented is employed in the analysis. Consequently, the mechanical and dielectric nonlinearities are included in the analysis as well as the semiconduction nonlinearity. Equations are derived for both the propagation velocity and the amplitude of the growing disturbance as a function of the state of the material immediately ahead of the wave front. These rather general results are specialized to the case of a homogeneous steady state ahead of the wave and the condition for the threshold field is determined. In this latter case the wave front propagates with constant velocity and the amplitude equation indicates the formation of a shock in a finite time for conditions conductive to domain formation. When the electrical conduction equation, wh...

On the viscous attenuation and velocity of acoustic surface waves in quartz

The viscous attenuation of surface waves propagating along various directions in some technologically important orientations of quartz has been obtained from a perturbation procedure which has been successfully employed in calculating the attenuation of such waves due to the finite electrical conductivity of overlaid thin metallic films and air loading. The attenuation and velocity curves show an interesting correlation which has a significant impact on the acoustic quality (Q) factor for such waves. Among other things the computational results indicate that a recently reported orientation (SST‐cut) which refers to the propagation direction 23° from the digonal axis of the BT‐cut possesses significantly lower viscous attenuation and beam spreading loss than the commonly used ST‐cut quartz for surface wave applications. The autocollimation property of surface waves in the new orientation is a consequence of the nature of the slowness curve in the neighborhood of the above mentioned propagation direction. T...

An analysis of transverse modes in acoustic surface wave resonators

A system of approximate surface wave equations employed in an earlier treatment of the reflection of straight‐crested surface waves by arrays of reflecting strips is extended to the case of variable‐crested surface waves. Although the basic straight‐crested surface wave velocities are determined as in the previous treatment, in the present case a reduction in straight‐crested surface wave velocity in the unplated region due to the adjacent plated regions, which is essential for the existence of the guided transverse modes, is determined by means of a perturbation procedure. The attendant depth dependence for each region is employed in the variational principle as in the earlier treatment, but now the variable cresting relation for the isotropic substrate is incorporated in the description. The resulting equations are applied in the determination of both the transverse modes in each region and the transmission line representation of each mode. The transverse wave numbers in a given mode are taken to be the...

Elastic surface waves guided by curved thin films

An approximation technique previously developed for straight elastic surface wave guides utilizing thin films is extended and applied to curved guides with large radius‐to‐guide‐width ratios. The dispersion curves for the circular guides are obtained by employing approximate bound modes in place of the actual wave functions, which are not fully bound. This procedure is quite accurate for the determination of dispersion curves in the propagation range of interest. Nevertheless, since the actual wave functions are not fully bound, some radiation is always present in a curved guide. This radiation loss accompanying the approximate bound modes is evaluated. A critical radius‐to‐guide‐width ratio, beyond which a calculation cannot be performed by the method employed, is defined, and it is shown that the radiation attenuation becomes prohibitively large even before this limiting ratio is reached. The problem of the lowest symmetric guided elastic surface wave in a straight guide incident on a curved guide in th...