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A Nonlinear FEM Model to Calculate Third-Order Harmonic and Intermodulation in TC-SAW Devices
(2018)
Nonlinearities in Temperature Compensated SAW (TC-SAW) devices in the 2 GHz range are investigated using a nonlinear finite element model by simultaneously considering both third-order intermodulation distortion (IMD3)and third harmonic (H3). In the employed perturbation approach, different contributions to the total H3, the direct and indirect contribution, are discussed. H3 and IMD3 measurements were fitted simultaneously using scaling factors for SiO 2 film and Cu electrode nonlinear material tensors in TC-SAW devices. We employ a P-Matrix simulation as intermediate step: Firstly, measurement and nonlinear P-Matrix calculations for finite devices are compared and coefficients of the P-Matrix simulation are determined. The nonlinear tensor data of the different materials involved in periodic nonlinear finite element method (FEM) computations are optimized to fit periodic P-Matrix calculations by introducing scaling factors. Thus, the contribution of different materials to the nonlinear behavior of TC-SAW devices is obtained and the role of materials is discussed.
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.
Recently a P-matrix and COM formalism was presented, which predicts third order intermodulation (IMD3) and triple beat with good accuracy and needs only a single nonlinearity constant. This formalism describes frequency dependence correctly. In this work the dependence of this nonlinearity constant on metalization ratio is investigated for aluminum metalization on LiTaO 3 (YXl)/42°. By comparison to test devices the nonlinearity constant is shown to be largely independent of metalization ratio. The nonlinear effect, however, strongly depends on metalization ratio, which is well described by the model. The linearity of a duplexer is optimized by reduction of metalization ratio and redesign of Tx branch topology.
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.
Existing ultrasonic stress evaluation methods utilize the acoustoelastic effect for bulk waves propagating in volume, which is unsuitable for a surface treated material, possessing a significant variation in material properties with depth. With knowledge of nonlinear elastic parameters – third-order elastic constants (TOEC) close to the surface of the sample, the acoustoelastic effect might be used with surface acoustic waves. This work is focused on the development of an independent method of TOEC measurement using the effect of nonlinear surface acoustic waves scattering – i.e. the effect of elastic waves interaction in a nonlinear medium.
In this paper, the possible three wave interactions of surface guided waves and bulk waves are described and formulae for the efficiency of harmonic generation and mode mixing are derived. A comparison of the efficiency of surface waves scattering in an isotropic medium for different interaction types is carried out with the help of nonlinear perturbation theory. First results for surface and bulk wave mixing with known second- and third-order elastic constants are shown.
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.
Elastic constants of components are usually determined by tensile tests in combination with ultrasonic experiments. However, these properties may change due to e.g. mechanical treatments or service conditions during their lifetime. Knowledge of the actual material parameters is key to the determination of quantities like residual stresses present in the medium. In this work the acoustic nonlinearity parameter (ANP) for surface acoustic waves is examined through the derivation of an evolution equation for the amplitude of the second harmonic. Given a certain depth profile of the third-order elastic constants, the dependence of the ANP with respect to the input frequency is determined and on the basis of these results, an appropriate inversion method is developed. This method is intended for the extraction of the depth dependence of the third-order elastic constants of the material from second-harmonic generation and guided wave mixing experiments, assuming that the change in the linear Rayleigh wave velocity is small. The latter assumption is supported by a 3D-FEM model study of a medium with randomly distributed microcracks as well as theoretical works on this topic in the literature.
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.
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.
In a recent paper it has been shown that the effective nonlinear constant which is used in a P-Matrix approach to describe third-order intermodulation (IMD3) in surface acoustic wave (SAW) devices can be obtained from finite element (FEM) calculations of a periodic cell using nonlinear tensor data [1]. In this paper we extend this FEM calculation and show that the IMD3 of an infinite periodic array of electrodes on a piezoelectric substrate can be directly simulated in the sagittal plane. This direct approach opens the way for a FEM based simulation of nonlinearities for finite and generalized structures avoiding the simplifications of phenomenological approaches.
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.
In numerical calculations, guided acoustic waves, localized in two spatial dimensions, have been shown to exist and their properties have been investigated in three different geometries, (i) a half-space consisting of two elastic media with a planar interface inclined to the common surface, (ii) a wedge made of two elastic media with a planar interface, and (iii) the free edge of an elastic layer between two quarter-spaces or two wedge-shaped pieces of a material with elastic properties and density differing from those of the intermediate layer.
For the special case of Poisson media forming systems (i) and (ii), the existence ranges of these 1D guided waves in parameter space have been determined and found to strongly depend on the inclination angle between surface and interface in case (i) and the wedge angle in case (ii). In a system of type (ii) made of two materials with strong acoustic mismatch and in systems of type (iii), leaky waves have been found with a high degree of spatial localization of the associated displacements, although the two materials constituting these structures are isotropic.
Both the fully guided and the leaky waves analyzed in this work could find applications in non-destructive evaluation of composite structures and should be accounted for in geophysical prospecting, for example.
A critical comparison is presented of the two computational approaches employed, namely a semi-analytical finite element scheme and a method based on an expansion of the displacement field in a double series of special functions.
Surface and interface acoustic waves are two-dimensionally guided waves, as their displacement field is plane-wave like regarding its dependence on the spatial coordinates parallel to the guiding plane, while it decays exponentially along the axis normal to that plane. When propagating at the planar surface or interface of homogeneous media, they are non-dispersive. Another type of non-dispersive acoustic waves which is, however, one-dimensionally guided, has displacement fields localized near the apex of a wedge made of an elastic material. In this short review, their propagation properties are described as well as theoretical and experimental methods which have been used for their analysis. Experimental findings are discussed in comparison with corresponding theoretical work and potential applications of this fascinating type of acoustic waves are presented.
A simple model is introduced that describes the interaction of surface acoustic waves (SAWs) with a 2D periodic array of objects on the surface that give rise to internal resonances. Such objects may be high-aspect ratio structures like micro-pillars fabricated of a material different from that of the substrate. The model allows for an approximate determination of the band structure for the acoustic modes in such systems. Results are presented for the dependence on structural parameters of a total bandgap in the non-radiative regime of a semi-infinite substrate, and it is shown how the frequency and radiation damping of vibrational modes can be determined that are associated with defects in the periodic 2D array.
Nonlinearity can give rise to intermodulation distortions in surface acoustic wave (SAW) devices operating at high input power levels. To understand such undesired effects, a finite element method (FEM) simulation model in combination with a perturbation theory is applied to find out the role of different materials and higher order nonlinear tensor data for the nonlinearities in such acoustic devices. At high power, the SAW devices containing metal, piezoelectric substrate, and temperature compensating (TC) layers are subject to complicated geometrical, material, and other nonlinearities. In this paper, third-order nonlinearities in TC-SAW devices are investigated. The materials used are LiNbO 3 -rot128YX as the substrate and copper electrodes covered with a SiO 2 film as the TC layer. An effective nonlinearity constant for a given system is determined by comparison of nonlinear P-matrix simulations to third-order intermodulation measurements of test filters in a first step. By employing these constants from different systems, i.e., different metallization ratios, in nonlinear periodic P-matrix simulations, a direct comparison to nonlinear periodic FEM-simulations yields scaling factors for the materials used. Thus, the contribution of the different materials to the nonlinear behavior of TC-SAW devices is obtained and the role of metal electrodes, substrate, and TC film are discussed in detail.
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.
The characteristic features and applications of linear and nonlinear guided elastic waves propagating along surfaces (2D) and wedges (1D) are discussed. Laser-based excitation, detection, or contact-free analysis of these guided waves with pump–probe methods are reviewed. Determination of material parameters by broadband surface acoustic waves (SAWs) and other applications in nondestructive evaluation (NDE) are considered. The realization of nonlinear SAWs in the form of solitary waves and as shock waves, used for the determination of the fracture strength, is described. The unique properties of dispersion-free wedge waves (WWs) propagating along homogeneous wedges and of dispersive wedge waves observed in the presence of wedge modifications such as tip truncation or coatings are outlined. Theoretical and experimental results on nonlinear wedge waves in isotropic and anisotropic solids are presented.