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Theoretical foundations of phononic crystals
Phononic crystals are periodic composite materials or structures which have elastic waves’ band gap property. So we can regard PnCs as the extension of the concept of crystals in solid state physics. The goal of the investigation of PnCs is essentially to study the propagation of elastic waves in periodic inhomogeneous media. Therefore, elasticity, as well as the lattice and band theories are the theoretical foundations of the investigation of PnCs.
In this chapter, firstly we will introduce some concepts and basic equations of dynamic elasticity. Then the propagation of surface acoustic waves on piezoelectric substrates will be studied, especially for the Rayleigh waves on lithium niobate semi-infinite substrate with different cut directions, considering with the boundary conditions. After that, theories of band structures and crystal lattice in solid state physics will be presented. Based on this, the PnC band gap mechanisms and properties will be summarized in this chapter. This chapter is served as a theoretical foundation of phononic crystal research.
Elasticity
Generalized Hooke’s Law
As we have already known, for one point inside the media, if there is an external force acting at this point, it will generate a stress and a strain, which are both second order tensors with 6 independent components. Under the ideal conditions, for one point inside the media, it should satisfy the linear relationship between stress and strain, as shown in Equation (2.1), which is called generalized Hooke’s Law.
In Equation (2.1), coefficients cmn (m, n = 1, 2, …, 6) are called elastic coefficients which form a tensor of 36 components, called elastic tensor (stiffnesss tensor).
If the object is made of inhomogeneous media, elastic coefficients are functions of coordinates x, y, z. Conversely, if the object is made of homogeneous media, it can be proved that [cmn] is a symmetric tensor with constants, with cmn = cnm (m, n = 1, 2, …, 6). Totally, there are 21 independent components. Specially, for isotropic homogeneous media, there are only 2 independent elastic coefficients. As a result, Equation (2.1) can be simplified in Equation (2.2).
Basic equations of elastodynamics
According to the theory of elastodynamics, for continuous isotropic homogeneous completely linear elastic material with micro deformation and without initial stress, we consider an arbitrarily small micro volume element (mass point). We can build three kinds of basic equations to describe the relationship among force, displacement, stress and strain, as shown below:
Differential equation of motion (expression of force and displacement), σ + ρf = ρü (2.6)
Geometrical equation (expression of displacement and strain), S ij = 1 (u + u ) (2.7) 2 i,j j,i
Physical equation (expression of stress and strain), σij = λθδij + 2μSij (2.8)
In Equation (2.6), fi is the external force acting on unit mass, ρ is the mass density of media, ui is the displacement, üi is the second order derivation of displacement with respect to time. The geometrical equation and the physical equation have been already mentioned above. Here for all equations, i, j = x, y, z.
In Cartesian coordinate system, there are 15 equations belonging to the three kinds of equations above, including 3 equations of motion, 6 geometrical equations and 6 physical equations with 15 unknowns, including 6 components of stress tensor, 6 components of strain tensor and 3 components of displacement ux, uy, uz. All of them are unknown functions of space coordinates x, y, z and time t. These 15 equations constitute a closed equation system, called basic equations of elastodynamics based on Cartesian coordinate system.
Among all the equations, not each of them includes all the unknown functions. So we can choose some of unknown functions as basic unknown functions, which can be solved firstly. Then on the basis of these solved basic unknown functions, all the others can be solved as well. As a result, we have 2 methods to solve this problem according to the choice of basic unknown function. For the first method, stress is chosen as the basic unknown function. For the second, displacement is chosen. In the equations of motion, there are partial derivative of stress and derivation of displacement-time, which is very hard to solve by the first method. So we always choose displacement as basic unknown functions to solve the equations of motion in elastodynamics. First, geometrical equation (2.7) is introduced into physical equation (2.8) to get the stress components expressed by displacement, which is introduced into equation of motion (2.6) for the second step. Finally the problem is solved. The equation of motion which chooses displacement as basic unknown function is called the Wiener Equation,
Surface acoustic waves on piezoelectric substrates
Because of the elasticity, every mass point in the media has an interaction with the adjacent mass points to reach an equilibrium state. If one point in the media has a disturbance, the relative positions among the adjacent points change. As a result, an elastic force between the disturbed point and the adjacent points appears. Therefore all the points around the disturbed points are into motion. This action is passed around to constitute a fluctuation started by a disturbance. We call this propagation process elastic wave. Since the dimension and size of the substrate where waves propagate can be different, the propagating properties are related to the imposed boundary conditions. Elastic waves can be divided into several types, such as bulk waves, surface acoustic waves (SAWs) and Lamb waves.
Table of contents :
Introduction
Chapter 1: State of the art
1.1 Historical context
1.2 Existing technologies
1.2.1 Simulation methods
1.2.2 Experimental study for microfabricated phononic crystals
1.3 Application of phononic crystals
1.3.1 Acoustic insulators
1.3.2 Acoustic diodes
1.3.3 Phononic crystal waveguides, cavities and filters
1.3.4 Thermal metamaterials and heat management
1.4 Motivation
Chapter 2: Theoretical foundations of phononic crystals
2.1 Elasticity
2.1.1 Generalized Hooke’s Law
2.1.2 Basic equations of elastodynamics
2.2 Surface acoustic waves on piezoelectric substrates
2.2.1 Plane waves
2.2.2 Surface acoustic waves
2.2.3 Rayleigh waves on LiNbO3 substrate with different cut directions
2.2.4 Piezoelectric consideration
2.2.5 Boundary conditions
2.3 Lattice and band theory
2.3.1 Lattice
2.3.2 Bloch Theorem
2.3.3 Band structures
2.4 PnC band gap mechanisms and properties
2.4.1 Bragg scattering
2.4.2 Local resonance
2.5 Conclusion
Chapter 3: Modeling of phononic crystals by the finite element method
3.1 Principles of finite element method
3.2 Modeling of band structures
3.3 Evidence of band gaps in pillar-based PnCs
3.4 Effect of pillar’s geometry
3.4.1 Effect of pillar’s height
3.4.2 Effect of pillar’s radius
3.4.3 Evolution of polarization by varying geometrical parameters
3.5 Effect of anisotropy
3.6 Material consideration
3.7 Temperature coefficients of frequency for surface modes
3.8 Calculation of transmission spectrum
3.8.1 Pillar-based PnCs with a square lattice on a Y128-X LiNbO3 substrate
3.8.2 Pillar-based PnCs with triangular lattices on a Y-Z LiNbO3 substrate
3.8.3 Discussion about Young’s Modulus
3.9 Conclusion
Chapter 4: Design, fabrication and characterization of pillar-based phononic crystals
4.1 Selection of materials
4.2 Device design
4.3 Fabrication of IDTs and PnCs
4.3.1 Substrate cleaning
4.3.2 Patterning by photolithography
4.3.3 Metallic deposition
4.3.4 Plasma-Enhanced Chemical Vapor Deposition (PECVD)
4.3.5 Etching
4.3.6 Electroplating of Nickel pillars
4.3.7 Procedure
4.4 Characterization
4.4.1PnCs morphology
4.4.2 Transmission spectrum measured by a vector network analyzer
4.5 Conclusion
Chapter 5: Ongoing work
5.1 Fabrication of 2D Si pillar-based phononic crystals
5.2 Self-assembly of magnetic nanoparticles
5.3 Fabrication of nickel nanowires by electroplating
5.4 Conclusion
Conclusion and Perspectives
Conclusion
Perspectives
Reference
