Layer Stack Analysis: Wave Propagation and Dispersion Diagram
Abstract
This lecture focuses on elastic wave propagation and layer stack dispersion diagram analysis for multilayer piezoelectric structures. First, the lecture reviews fundamentals of acoustic wave propagation in layered media commonly used in Bulk Acoustic Wave (BAW), Film Bulk Acoustic Wave Resonator (FBAR), and Surface Acoustic Wave (SAW) devices. Next, the lecture derives and discusses wave propagation solutions for infinite media, half-spaces, finite-thickness layers, and arbitrary multilayer stacks. Furthermore, the lecture considers numerical techniques and computational algorithms required for dispersion diagram evaluation. Finally, the lecture analyzes and compares dispersion diagram properties for various practical materials and multilayer structures.
Elastic Waves in Piezoelectric Solids
First, the lecture reviews basic properties of elastic waves in solids and discusses classification of acoustic waves including Bulk Acoustic Waves (BAW) and Surface Acoustic Waves (SAW). Furthermore, the lecture considers elastic, piezoelectric, and dielectric material constants as well as matrix symmetry properties of piezoelectric crystals. In particular, the lecture pays special attention to Aluminium Nitride (AlN) and its applications in modern BAW and FBAR devices.
Wave Propagation Problem and Mathematical Formulation
Next, the lecture formulates the general wave propagation problem in anisotropic piezoelectric media. The lecture derives a system of ordinary differential equations (ODEs) describing elastic wave propagation. Furthermore, Christoffel tensors and matrices are introduced to simplify and unify the notation. As a result, the original wave propagation problem reduces to standard or generalized eigenvalue problems. The lecture then discusses eigenvalues, eigenvectors, and their physical interpretation. In addition, the lecture derives the Christoffel secular equation and considers its solution. Moreover, the lecture discusses dyadic Green’s function formalism and its applications.
Bulk and Surface Acoustic Wave Solutions
The lecture then considers wave propagation in infinite media, half-spaces, and finite-thickness layers. Moreover, the lecture introduces surface and interfacial boundary conditions for multilayer structures. In addition, partial-wave representation and general solutions of the acoustic wave problem are discussed. For finite-thickness layers, the lecture derives the layer transfer matrix and reviews the Fahmy–Adler solution of the multilayer wave propagation problem.
Multilayer Stack Analysis and Dispersion Diagram Calculation
Furthermore, the lecture generalizes wave propagation analysis to arbitrary multilayer stacks by recurrent calculation of the overall transfer matrix. Then, the lecture applies electromechanical interfacial and terminal boundary conditions to derive the complete multilayer solution. Based on the transfer-matrix formalism, the lecture develops an algorithm for dispersion diagram calculation. Finally, the lecture analyzes numerical instabilities of the transfer-matrix method and compares them with state-of-the-art techniques providing improved numerical stability.
Dispersion Diagram Analysis and Applications
Finally, the lecture discusses physical interpretation of multilayer dispersion diagrams and explains their practical applications in acoustic wave device analysis and design. Furthermore, tutorial examples illustrating dispersion diagram calculation and interpretation are presented. In particular, the lecture demonstrates analysis of FBAR dispersion diagrams and discusses relationships between dispersion characteristics, acoustic modes, and resonator performance.
Throughout the lecture, the material is illustrated by the author's proprietary MATLAB® dispersion diagram analysis software along with a live computer demonstration.