The Main Lateral Mode Approximation of a Film Bulk Acoustic Resonator With Perfect Metal Electrodes.

Using first principles and the constitutive equations of a piezoelectric crystal, we solve the 2-D problem inside a three-layer film bulk acoustic resonator (FBAR) in order to study the dispersion and parasitic lateral modes' characteristic of the structure. In our main lateral mode approximation, described here in detail, we construct the acoustic wave by combining the ideal "piston" mode and the main dispersion branch lateral mode. By limiting our analysis to the practical range of frequencies near the series resonance of the stack, where the lateral component $k_{x}$ of the ${k}$ vector is small, we find analytical expressions for the FBAR acoustic wave and for the dispersion of the three-layer stack. When lateral boundary conditions are added to the acoustic problem of a laterally finite resonator, we employ our theory to estimate the amplitude and the propagation of the lateral modes and then compare the theoretical predictions with the measurements of fabricated FBARs and finite-element simulation results. We are able to distinguish between a "clamped" and a "quasi-free" lateral interface by comparing the amplitude strength of the lateral modes produced, and we discuss how optimum lateral boundaries can be engineered with perimeter frames for realistic resonators.

An Investigation of Lateral Modes in FBAR Resonators.

Using first principles and the constitutive equations for a piezoelectric, we solve the 2D acoustic wave inside a single, infinite, piezoelectric membrane in order to study the dispersion of Thin Film Bulk Acoustic Resonator (FBAR) lateral modes, with and without infinitely-thin electrodes. The acoustic eigenfunction is a dual wave, composed of longitudinal and shear components, able to satisfy the 2D acoustic boundary conditions at the vacuum interfaces. For the single piezoelectric slab we obtain analytical expressions of the dispersion for frequencies near the longitudinal resonant frequency (Fs) of the resonator. These expressions are more useful for the understanding of dispersion in FBARs and more elegant than numerical methods like Finite Element Modeling (FEM) and various matrix methods. We additionally find that the interaction between the resonator's electrodes and the acoustic wave modifies the lateral mode dispersion when compared to the case with no electrodes. When correctly accounting for these interactions the dispersion zero is placed clearly at Fs, unlike what is calculated from a 2D model without electrodes where the dispersion zero is placed at Fp. This is important since all experimental evidence of measures FBAR resonators shows that the dispersion zero is at Fs. Furthermore, we introduce an electrical current flow model for the propagating acoustic wave inside the electroded piezoelectric and based on this model we can discuss an electrode-loss mechanism for FBAR lateral modes which depends on dispersion. From our model it results that lateral modes with real kx have higher electrode dissipation if they are closer to the resonant frequency. This is consistent with the typical behavior of measured FBAR filters where the maximum lateral mode damage on the insertion loss takes place for frequencies immediately below Fs.

Finite-Element Analysis of Bulk-Acoustic-Wave Devices: A Review of Model Setup and Applications.

In this paper, the principles of finite-element modeling for the electroacoustic simulation of bulk-acoustic-wave devices will be summarized. We will outline the model setup including governing equations and boundary conditions, as well as its efficient computer implementation. Particular emphasis will be given to tailoring the model dimension to the specific requirements of the desired investigation. As 3-D simulations still require a major effort, it will be illustrated that various aspects of device physics and design can already be addressed by fast and efficient 2-D simulations. Multiple theoretical and experimental evidence will be presented to demonstrate the validity of the modeling concepts. Based on various examples, it will be sketched how to benefit from numerical simulations for understanding fundamental effects, designing devices for actual products, and exploring novel technologies.

Experimental investigation of acoustic substrate losses in 1850-MHz thin film BAW resonators.

After optimizing for electromechanical coupling coefficient K(2), the main performance improvement in the thin film bulk acoustic wave resonators and filters can be achieved by improving the Q value, i.e., minimizing the losses. In Bragg-reflector-based solidly mounted resonator technology, a significant improvement of Q has been achieved by optimizing the reflector not only for longitudinal wave, the intended operation mode, but also for shear waves. We have investigated the remaining acoustic radiation losses to the substrate in so-optimized 1850-MHz AlN resonators by removing the substrate underneath the resonators and comparing the devices with and without substrate by electrical characterization before and after the substrate removal. Several methods to extract Q-values of the resonators are compared. Changes caused by substrate removal are observed in resonator behavior, but no significant improvement in Q-values can be confirmed. Loss mechanisms other than substrate leakage are concluded to dominate the resonator Q-value. Difficulties of detecting small changes in the Q-values of the resonators are also discussed.

Virtual optical experiments. Part I. Modeling the measurement process.

In recent years, internal laser probing techniques that exploit the electro-optical and the thermo-optical effects have been introduced. Space-resolved and time-resolved measurements of charge-carrier and temperature distributions in the interior of semiconductor samples have thus become possible. For a profound analysis and the optimization of these measurement techniques, a physically rigorous model for simulating the entire measurement process is presented. The model includes the electrothermal device simulation of the sample's operating condition, the calculation of the resulting refractive-index modulations, the simulation of wave propagation through the device under test, the imaging lenses and aperture holes, and the simulation of the detector response. As an essential part of this model, a numerically efficient algorithm for simulating wave propagation in large computational domains has been developed. The decisive step is introduction of a suitably chosen set of computational variables that allows a significantly coarser discretization width without loss of accuracy.

Virtual optical experiments. Part II. Design of experiments.

In Part I of this study [J. Opt. Soc. Am. A 20, 698 (2003)], we presented a physically rigorous model for simulating optical probing techniques. We now introduce the concept of virtual experiments as the fundamental strategy for analyzing the measurement techniques and for supporting the design of the experiments. Thus a theoretical study of parasitic effects, the accuracy of the experiment, and the optimum probing conditions becomes possible. In our first example of application, free-carrier absorption measurements are discussed. We present quantitative results for the optimum sample geometry and the optimum optical setup. In addition, we demonstrate that backside laser probing, a typical representative of interferometric techniques, provides excellent spatial resolution and constitutes a powerful method to detect hot spots in the investigated sample.