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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.
In this work a set of nonlinear coupled COM equations at interacting frequencies is derived on the basis of nonlinear electro-elasticity. The formalism is presented with the aim of describing intermodulation distortion of third-order (IMD3) and triple beat. The resulting COM equations are translated to the P-matrix formalism, where care is taken to obtain the correct frequency dependence. The scheme depends on two frequency-independent constants for an effective third-order nonlinearity. One of these two constants is negligibly small in the systems considered here. The P-matrix approach is applied to single filters and duplexers on LiTaO 3 (YXl)/42° operating in different frequency ranges. Both IMD3 and triple beat show good agreement with measurement.
The growing complexity in RF front-ends, which support carrier aggregation and a growing number of frequency bands, leads to tightened nonlinearity requirements in all sub-components. The generation of third order intermodulation products (IMD3) are typical problems caused by the non-linearity of SAW devices. In the present work, we investigate temperature compensating (TC) SAW devices on Lithium Niobate-rot128YX. An accurate FEM simulation model [1] is employed, which allows to better understand the origin of nonlinearities in such acoustic devices.
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.
The existence of acoustic waves with displacements localized at the tip of an isotropic elastic wedge was rigorously proven by Kamotskii, Zavorokhin and Nazarov. This proof, which is based on a variational approach, is extended to rectangular anisotropic wedges. For two high-symmetry configurations of rectangular edges in elastic media with tetragonal symmetry, a criterion is derived that allows identifying the boundary between the regions of existence for wedge modes of even and odd symmetry in regions of parameter space, where even- and odd-symmetry modes do not exist simultaneously. Furthermore, rectangular edges with non-equivalent surfaces are analyzed, and it is shown that at rectangular edges of cubic elastic media with one (110) surface and one (001) surface, a tip-localized guided wave always exists, apart from special cases that are characterized.
Laser pulses focused near the tip of an elastic wedge generate acoustic waves guided at its apex. The shapes of the acoustic wedge wave pulses depend on the energy and the profile of the exciting laser pulse and on the anisotropy of the elastic medium the wedge is made of. Expressions for the acoustic pulse shapes have been derived in terms of the modal displacement fields of wedge waves for laser excitation in the thermo-elastic regime and for excitation via a pressure pulse exerted on the surface. The physical quantity considered is the local inclination of a surface of the wedge, which is measured optically by laser-probe-beam deflection. Experimental results on pulse shapes in the thermo-elastic regime are presented and confirmed by numerical calculations. They pertain to an isotropic sharp-angle wedge with two wedge-wave branches and to a non-reciprocity phenomenon at rectangular silicon edges.
Among the various types of guided acoustic waves, acoustic wedge waves are non-diffractive and non-dispersive. Both properties make them susceptible to nonlinear effects. Investigations have recently been focused on effects of second-order nonlinearity in connection with anisotropy. The current status of these investigations is reviewed in the context of earlier work on nonlinear properties of two-dimensional guided acoustic waves, in particular surface waves. The role of weak dispersion, leading to solitary waves, is also discussed. For anti-symmetric flexural wedge waves propagating in isotropic media or in anisotropic media with reflection symmetry with respect to the wedge’s mid-plane, an evolution equation is derived that accounts for an effective third-order nonlinearity of acoustic wedge waves. For the kernel functions occurring in the nonlinear terms of this equation, expressions in terms of overlap integrals with Laguerre functions are provided, which allow for their quantitative numerical evaluation. First numerical results for the efficiency of third-harmonic generation of flexural wedge waves are presented.
For an elastic medium containing a homogeneous distribution of micro-cracks, an effective one-dimensional stress-strain relation has been determined with finite element simulations. In addition to flat micro-cracks, voids were considered that contain a Hertzian contact, which represents an example for micro-cracks with internal structure. The orientation of both types of micro-cracks was fully aligned or, for flat micro-cracks, totally random. For micro-cracks with Hertzian contacts, the case of random orientation was treated in an approximate way. The two types of defects were found to give rise to different degrees of non-analytic behavior of the effective stress-strain relation, which governs the nonlinear propagation of symmetric (S0) Lamb waves in the long-wavelength limit. The presence of flat micro-cracks causes even harmonics to grow linearly with propagation distance with amplitudes proportional to the amplitude of the fundamental wave, and gives rise to a static strain. The presence of the second type of defects leads to a linear growth of all harmonics with amplitudes proportional to the power 3/2 of the fundamental amplitude, and to a strain-dependent velocity shift. Simple expressions are given for the growth rates of higher harmonics of S0 Lamb waves in terms of the parameters occurring in the effective stress-strain relation. They have partly been determined quantitatively with the help of the FEM results for different micro-crack concentrations.
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.
Micro-cracks give rise to non-analytic behavior of the stress-strain relation. For the case of a homogeneous spatial distribution of aligned flat micro-cracks, the influence of this property of the stress-strain relation on harmonic generation is analyzed for Rayleigh waves and for acoustic wedge waves with the help of a simple micromechanical model adopted from the literature. For the efficiencies of harmonic generation of these guided waves, explicit expressions are derived in terms of the corresponding linear wave fields. The initial growth rates of the second harmonic, i.e., the acoustic nonlinearity parameter, has been evaluated numerically for steel as matrix material. The growth rate of the second harmonic of Rayleigh waves has also been determined for microcrack distributions with random orientation, using a model expression for the strain energy in terms of strain invariants known in a geophysical context.