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In the framework of electro-elasticity theory and the finite element method (FEM), a model is set up for the computation of quantities in surface acoustic wave (SAW) devices accounting for nonlinear effects. These include second-order and third-order intermodulations, second and third harmonic generation and the influence of electro-acoustic nonlinearity on the frequency characteristics of SAW resonators. The model is based on perturbation theory, and requires input material constants, e.g., the elastic moduli up to fourth order for all materials involved. The model is two-dimensional, corresponding to an infinite aperture, but all three Cartesian components of the displacement and electrical fields are accounted for. The first version of the model pertains to an infinite periodic arrangement of electrodes. It is subsequently generalized to systems with a finite number of electrodes. For the latter version, a recursive algorithm is presented which is related to the cascading scheme of Plessky and Koskela and strongly reduces computation time and memory requirements. The model is applied to TC-SAW systems with copper electrodes buried in an oxide film on a LiNbO3 substrate. Results of computations are presented for the electrical current due to third-order intermodulations and the displacement field associated with the second harmonic and second-order intermodulations, generated by monochromatic input tones. The scope of this review is limited to methodological aspects with the goal to enable calculations of nonlinear quantities in SAW devices on inexpensive and easily accessible computing platforms.
In this work the nonlinear behavior of layered surface acoustic wave (SAW) resonators is studied with the help of finite element (FE) computations. The full calculations depend strongly on the availability of accurate tensor data. While there are accurate material data for linear computations, the complete sets of higher-order material constants, needed for nonlinear simulations, are still not available for relevant materials. To overcome this problem, scaling factors were used for each available nonlinear tensor. The approach here considers piezoelectricity, dielectricity, electrostriction, and elasticity constants up to the fourth order. These factors act as a phenomenological estimate for incomplete tensor data. Since no set of fourth-order material constants for LiTaO3 is available, an isotropic approximation for the fourth-order elastic constants was applied. As a result, it was found that the fourth-order elastic tensor is dominated by one-fourth order Lamé constant. With the help of the FE model, derived in two different, but equivalent ways, we investigate the nonlinear behavior of a SAW resonator with a layered material stack. The focus was set to third-order nonlinearity. Accordingly, the modeling approach is validated using measurements of third-order effects in test resonators. In addition, the acoustic field distribution is analyzed.
The laser ultrasound (LU) technique has been used to determine dispersion curves for surface acoustic waves (SAW) propagating in AlScN/Al2O3 systems. Polar and non-polar Al0.77Sc0.23N thin films were prepared by magnetron sputter epitaxy on Al2O3 substrates and coated with a metal layer. SAW dispersion curves have been measured for various propagation directions on the surface. This is easily achieved in LU measurements since no additional surface structures need to be fabricated, which would be required if elastic properties are determined with the help of SAW resonators. Variation of the propagation direction allows for efficient use of the system’s anisotropy when extracting information on elastic properties. This helps to overcome the complexity caused by a large number of elastic constants in the film material. An analysis of the sensitivity of the SAW phase velocities (with respect to the elastic moduli and their dependence on SAW propagation direction) reveals that the non-polar AlScN films are particularly well suited for the extraction of elastic film properties. Good agreement is found between experiment and theoretical predictions, validating LU as a non-destructive and fast technique for the determination of elastic constants of piezoelectric thin films.
This work focuses on the dependencies between typical design parameters of surface acoustic wave (SAW) resonators and the nonlinear emitted signals of second and third order. The parameters metalization ratio and pitch are used as examples, but the approach can be extended to other design parameters as well. It is shown, that the interaction between the nonlinear current generation and the linear admittance is defining the measured nonlinear power signals. It is also discussed, that changes in linear properties get more pronounced in nonlinear responses. Therefore, slight effects on linear parameters will have significant influence on the observed nonlinearity.
Surface treatment intensity monitoring is still an open and challenging nondestructive testing problem. For the estimation of residual stress with ultrasonic measurements, local linear and nonlinear elastic constants are needed as input. In this paper, nonlinear elastic-wave interactions (also called wave mixing or scattering) — namely, the generation of secondary ultrasonic waves in a nonlinear medium — are considered as a prospective means for near-surface nonlinear elastic parameter evaluation. The allowed interactions between bulk and surface waves, as well as the dependence of the scattering efficiency on the frequency and angle between source waves, were investigated through an analytical model, then compared with FEM simulations and experimental results. Finally, possible future steps for the development of the applied methods for the determination of near-surface higher-order elastic constants are discussed. In addition, several problem-relevant data processing procedures are presented.
Acoustic waves are investigated which are guided at the edge (apex line) of a wedge-shaped elastic body or at the edge of an elastic plate. The edges contain a periodic sequence of modifications, consisting either of indentations or inclusions with a different elastic material which gives rise to high acoustic mismatch. Dispersion relations are computed with the help of the finite element method. They exhibit zero-group velocity points on the dispersion branches of edge-localized acoustic modes. These special points also occur at Bloch-Floquet wavenumbers away from the Brillouin zone boundary. Deep indentations lead to flat branches corresponding to largely non-interacting, Einstein-oscillator like vibrations of the tongues between the grooves of the periodic structure. Due to the nonlinearity of the elastic media, quantified by their third-order elastic constants, an acoustic mode localized at a periodically modified edge generates a second harmonic which partly consists of surface and plate modes propagating into the elastic medium in the direction vertical to the edge. This acoustic radiation at the second-harmonic frequency is investigated for an elastic plate and a truncated sharp-angle wedge with periodic inclusions at their edges. Unlike nonlinear bulk wave generation by surface acoustic waves in an interdigital structure, surface and plate mode radiation by edge-localized modes can be visualized directly in laser-ultrasound experiments.
Nonlinear acoustic waves are considered that have displacements localized at the tip of an elastic wedge. The evolution equation governing their propagation is discussed and compared with its analogues pertaining to nonlinear acoustic surface and bulk waves. Solitary wave solutions of the evolution equation have been determined numerically for the cases of two rectangular edges which may be viewed as generated by splitting a half-space, consisting of crystalline silicon, into two quarter-spaces. For these two geometries, the kernel in the nonlinear terms of the evolution equation has been calculated from the second-order and third-order elastic constants of silicon, and weak dispersion due to tip truncation has been considered. Solitary pulse shapes have been computed and collisions of solitary pulses have been simulated for various relative speeds of the two collision partners. Collision scenarios for the two wedge geometries were found to differ considerably. Special attention is paid to the peculiar interaction of two initially identical solitary pulses.
Surface acoustic waves are propagated toward the edge of an anisotropic elastic medium (a silicon crystal), which supports leaky waves with a high degree of localization at the tip of the edge. At an angle of incidence corresponding to phase matching with this leaky wedge wave, a sharp peak in the reflection coefficient of the surface wave was found. This anomalous reflection is associated with efficient excitation of the leaky wedge wave. In laser ultrasound experiments, surface acoustic wave pulses were excited and their reflection from the edge of the sample and their partial conversion into leaky wedge wave pulses was observed by optical probe-beam deflection. The reflection scenario and the pulse shapes of the surface and wedge-localized guided waves, including the evolution of the acoustic pulse traveling along the edge, have been confirmed in detail by numerical simulations.
Increasing power density causes increased self-generation of harmonics and intermodulation. As this leads to violations of the strict linearity requirements, especially for carrier aggregation (CA), the nonlinearity must be considered in the design process of RF devices. This raises the demand of accurate simulation models. Linear and nonlinear P-Matrix/COM models are used during the design due to their fast simulation times and accurate results. However, the finite element method (FEM) is useful to get a deeper insight in the device's nonlinearities, as the total field distributions can be visualized. The FE method requires complete sets of material tensors, which are unknown for most relevant materials in nonlinear micro-acoustics. In this work, we perform nonlinear FEM simulations, which allow the calculation of nonlinear field distributions of a lithium tantalate based layered SAW system up to third order. We aim at achieving good correspondence to measured data and determine the contributions of each material layer to the nonlinear signals. Therefore, we use approximations circumventing the issue of limited higher order tensor data. Experimental data for the third order nonlinearity is shown to validate the presented approach.
Laser ultrasound was used to determine dispersion curves of surface acoustic waves on a Si (001) surface covered by AlScN films with a scandium content between 0 and 41%. By including off-symmetry directions for wavevectors, all five independent elastic constants of the film were extracted from the measurements. Results for their dependence on the Sc content are presented and compared to corresponding data in the literature, obtained by alternative experimental methods or by ab-initio calculations.