Contents of the Chapters

There are six chapters in this thesis. Chapter 1 is the introduction, with the research motivation and literature review. In Chapter 2 we show the method to calculate propa-gating eigenmodes in piezoelectric substrate covered with a phononic guiding layer, with a detailed discussion of their band structures. Chapter 3 is the design method of the one-port Love wave resonator. Also the physics of Love wave incident to the PC is studied. Chapter 4 is the fabrications of the experiment devices. The process of how the devices were made are shown in detail. Chapter 5 shows the experiment results with discussion. And Chapter 6 is the conclusions and future works.

Chapter 2

Love Wave Dispersions of a Phononic SiO ₂ /Quartz Layered Structure

In this chapter, the numerical methods for calculating the Love wave dispersions in three different structures are proposed. They are (1) CVD SiO2/quartz layered piezoe-lectric half space, (2) CVD SiO2/aluminum electrodes grating/quartz structure, with consideration of the electrodes’ mechanical loading effect, which is a 1-D periodic structure, and (3) phononic SiO2/quartz, a 2-D periodic structure.

First the governing equations of the “quasi-static” approximated model of piezoe-lectric solid are derived, and then the theory of waves in periodic structure is introduced.

After we have the governing equations and boundary conditions the finite element method is employed to solve the differential equations to find the dispersion relation of the 3 structures.

2.1 The Mathematical Model of Acoustic Waves in Piezo-electric solids

2.1.1 Piezoelectric acoustic wave equations

The mechanical system and the electrical system of the piezoelectric solid are cou-pled together in their constitutive relations. Meanwhile, Newton’s 2^nd law of motion

∇ ⋅T + X = ρ ∂²u

∂t² , (2.1)

and the Maxwell equations

∇⋅D = ρ^f (2.2)

are still satisfied, where T, X, and u are the Cauchy stress tensor, the body force density vector, and the displacement vector, respectively, while D, B, E, H, _ρf,and _Jfare the electric displacement field, the magnetic flux density, the electric field, the magnetic field strength, the free charge density, and the free current density, respectively. To simplify the mathematical model, some fair assumptions are made as the following.

Body forces such as gravity are ignored ( _{X = 0}). The piezoelectric material is as-sumed to be dielectric, i.e., no free charge or free current exist ( _ρf = 0, J_f = 0). And the

“quasi-static” approximation is introduced, that is, the electric field and the magnetic field are assumed to be static (

∂B

∂t = 0, ∂D

∂t = 0). For the reason that we are focusing on the “quasi-acoustic waves,” and the frequencies of the “quasi-electromagnetic waves”

are much higher under the same wavenumber so that the electromagnetic field can be assumed to be static. Therefore the electric field and the magnetic field become decou-pled; moreover, the magnetic field is decoupled from the piezoelectric system. Thus the governing equations of the piezoelectric solid are reduced to the following two equa-tions (with indicial notation)

T_{ij, j}=ρ!!u_i, (2.6)

D_i,i= 0, (2.7)

with the constitutive equations which couple the two systems together:

T_ij=c_ijkl^E S_kl−e_kijE_k, (2.8) represent the elastic stiffness tensor under a constant electric field, the permittivity ten-sor with a constant strain, and the piezoelectric coupling tenten-sor, respectively. Substitut-ing equations (2.8) and (2.9) into equations (2.6) and (2.7), with the electric field-potential relation E_i= −φ_,i, and the strain-displacement relation

S_kl= 1

2(u_k,l+u_l,k), we obtain the governing equations of piezoelectricity with variables

u and φ,

(

c_ijkl^E u_k,l+e_kijφ_,k

)

_{, j} ^{=ρ!!ui, (2.10)}

(

e_iklu_k,l−ε_ij^Sφ_{, j}

)

_,k^{= 0. (2.11)}

In the case of time harmonic motion (when calculating propagating eigenmodes or simulating the frequency response of a device), the displacement vector and the electric

potential can be represented as

u(x, y,z,t) = ˆu(x, y,z)e^iωt, (2.12) φ(x, y,z,t) = ˆφ(x, y,z)e^iωt, (2.13) where ω is the angular frequency. Hence equations (2.10) and (2.11) can be rewritten as

(

c_ijkl^E ˆu_k,l+e_kijˆφ_,k

)

, j= −ρω²ˆu_i, (2.14)

(

e_iklˆu_k,l−ε_ij^Sˆφ_{, j}

)

_,k^{= 0. (2.15)}

With proper boundary conditions (BCs), the solutions of u and φ can be found. In this thesis, the rest jobs of solving the differential equations are leaved to a commercial fi-nite element method (FEM) software COMSOL Multiphysics.

2.1.2 Coordinate transformation for material constants

All piezoelectric materials are anisotropic; their properties depend on the frame of reference, i.e., the crystal cut and the orientation of wave propagation, so they should be first correctly obtained. Usually the material properties with respect to (w.r.t.) the crys-tal axes (denoted by X, Y, Z, defined by cryscrys-tal symmetry) are found in books. As for some commonly used material with specific crystal cuts such as ST-cut quartz, 128°-YX lithium niobate, the material properties can also be found in literatures. How-ever with different definitions of the specimen axes, they are easily confused and mis-used. In this thesis, the specimen axes (x, y, z) are defined as follows. The axis normal to the substrate’s top surface is defined as the z-axis, with the wave propagating direc-tion defined as the x-axis, the y-axis is then obtained by the right-hand rule, as shown in Fig. 2.1. According to Euler’s rotation theorem, any rotation can be decomposed into three successive rotations with three angles (Φ, Θ, Ψ), w.r.t. the axes in a certain order.

However the decomposition is not unique and is order-dependent. In the thesis the

Z-X’-Z” convention is adopted, that is, the specimen axes is rotated from the crystal ax-is first with an angle Φ w.r.t. the Z-axax-is, then rotated with an angle Θ w.r.t. the current

x-axis (the X’-axis), and finally rotated with an angle Ψ w.r.t. the current z-axis (the Z”-axis). Namely, a rotation rule (ZXtlt) Φ/Θ/Ψ following IEEE standard on

piezoelec-tricity [39], is applied, with the plate length direction coincident with the wave propa-gating orientation.

Let Vi and V’j denote the components of a vector V w.r.t. the crystal coordinates and the specimen coordinates respectively, there exist a transformation rule between Vi

and V’j,

Similarly, the permittivity, the piezoelectric coupling tensor, and the elastic stiffness tensor, which are tensors of order 2, 3, and 4, respectively, have the transformation rules as

To support Love wave propagation, it is important choosing the material combina-tion of the layered half-space structure. First the SH wave velocity in a virtual un-bounded medium of the layer material must be slower than that of the substrate material, for it is the condition of existence of Love waves. [40], [41] Second, the substrate

mate-rial (with specific crystal cut) should have strong piezoelectric coupling to SH type wave (SH bulk wave, SSBW, SH LSAW, or BGS wave), but with zero piezoelectric coupling to Rayleigh modes or other (P or SV) bulk wave modes to prevent the spurious response. Several crystal cuts of lithium niobate (LN, LiNbO3), lithium tantalate (LT, LiTaO3) and quartz are found desirable for such requirements. [42] Eventually, 90°-z-rotated ST-cut quartz (henceforth 90ST quartz) was chosen for there is a fast (5000m/s) SSBW on it and ST-cut quartz is commercially available with a lower price than LT or LN. But there is a disadvantage that is the poor piezoelectric coupling of quartz. On the other hand, the layer material was chosen to be amorphous silicon diox-ide (SiO2, silica henceforth), which is the most frequently used dielectric material in in-tegrated circuit (IC) manufacturing hence the deposition and etching techniques for sil-ica are well developed. In this study the silsil-ica was deposited by plasma enhanced chemical vapor deposition (PECVD) method. Its shear wave velocity is estimated to be 3438 m/s.

The elastic constants of quartz used in this study were from [43], with the kind suggestion from Prof. Yook-Kong Yong. The properties of quartz with respect to its crystal axes are listed in Table 2.1. The ST-cut quartz, in words of wafer manufacturers, is a singly rotated (42°45’ along X-axis) Y-cut quartz and the corresponding Euler an-gles (defined in the previous subsection) are _Φ,Θ,Ψ

( )

= 0,132°45', 0

( )

. As for the

“90°-z-rotated ST-cut quartz,” it is physically no difference from a ST-cut quartz sub-strate, but with the wave propagation direction parallel to the y-axis of ST-cut quartz, and perpendicular to the X-axis of quartz (or the x-axis of ST-cut quartz). So actually we demand ST-cut quartz wafer from the supplier, and make a 90°-rotated layout. The Euler angles of 90ST quartz are

(

Φ Θ Ψ =^{, ,}

) (

0,132 45', 90° °

)

. The properties of ST-cut

quartz and 90°-rotated ST-cut quartz w.r.t. their specimen axes are detailed in Table 2.2 and Table 2.3, respectively. It can be seen that the difference between them is that the components related to axes x or y are exchanged. ST-cut quartz is usually used as a Rayleigh wave device, while on a 90ST quartz SH type SSBW can be generated but Rayleigh wave is decoupled with the applied electric field from the interdigital trans-ducer (IDT). A schematic drawing is shown in Fig. 2.3.

The amorphous silica layer in this study is deposited by a plasma-enhanced chem-ical vapor deposition (PECVD) method, which is described in Chapter 4. The elastic properties are listed in Table 2.4, where the elastic stiffness constants were obtained by a SAW method. By measuring the Love wave and Rayleigh wave velocities on the sili-ca film/quartz substrate, the two elastic constants sili-can be obtained. The method is de-tailed in the Appendix . It shows a significant difference from the bulk SiO2 (fused quartz).

2.2 Waves in periodic structures

2.2.1 Direct and reciprocal lattices

Considering a 1-D, 2-D or 3D lattice, a basis {a}, {a1, a2} or {a1, a2, a3} can be chosen by selecting a lattice point as the origin, and the nearest points define the basis vector, such that the vector coordinate of any lattice points can be given by linear com-binations of the basis with integer coefficients. Note that the basis is not unique. A 2-D case is shown in Fig 2.4, where {a1, a2} or {a’1, a’2} both can be the basis but here the grid lines are drawn w.r.t {a1, a2}.

A “primitive unit cell” can be used to represent the lattice, as it is the smallest per-iod of the lattice. Various primitive unit cells can be chosen but the Wigner-Seitz cell is

the most common and maybe the most useful unit cell. The Wigner-Seitz cell can be chosen by constructing a domain (an interval, a polygon, or a polyhedron stand for 1-D, 2-D or 3-D lattices, respectively) about a lattice point. The boundaries (end points, bor-derlines or faces) of the cell are points, lines of planes that are perpendicular bisectors of the lines joining the origin with neighboring lattice points [44]. A Wigner-Seitz cell in the 2-D case is shown schematically in Fig 2.4.

Until here the discussion is about the lattice defined by {a}, {a1, a2} or {a1, a2, a3} basis. We refer to it as the direct lattice. For each lattice we may define a reciprocal

lat-tice that has basis {b}, {b

1, b2} or {b1, b2, b3} given by the equation scaling factor (2π)⁻¹). To find the basis of the reciprocal lattice, we first build two 2×2 (3×3) matrices containing the Cartesian components of the bases {a1, a2} and {b1, b2}

Where I is the identity matrix, therefore it follows that

B = 2πA⁻¹. (2.24) Hence the basis of the reciprocal lattice is obtained from a given direct lattice. And the reciprocal lattice in 1-D case can be simply represented as

b = 2π a .

The wave vector (k) for a wave propagating in the direct lattice is drawn in the re-ciprocal lattice. The direct lattice gives the periodicity of the medium (x-space), while the reciprocal lattice gives the periodicity of the frequency of the waves propagating through the medium (k-space) [44].

Since the reciprocal lattice shows the periodicity in k-space, just like the direct lat-tice in x-space, a primitive unit cell can be found to describe the whole reciprocal latlat-tice.

That is the first Brillouin zone, obtained by constructing a Wigner-Seitz zone in the re-ciprocal lattice. It is worth noting that discontinuous frequency spectrum often occurs on the boundary of the first Brillouin zone that can be realized as Bragg’s reflection condition

k ⋅a = nπ, (2.25) where n is an integer. Or the level repulsion due to coupling with diffracted waves.

Fig 2.5 shows the direct lattice and the reciprocal lattice of a 2-D square lattice, as

well as the Wigner-Seitz cell of the direct lattice and the first Brillouin zone in the re-ciprocal lattice. Some special points Γ, X, M, and Y, on the boundary of the Brillouin zone are labeled. And the irreducible Brillouin zone is the first Brillouin zone reduced by all of the symmetries of the lattice. The irreducible Brillouin zone of a square lattice consists of points in the space, is the triangular gridded by the three points Γ, X, and M.

But if we consider lattices containing anisotropic materials, the irreducible Brillouin zones may be different. In this study the lattice is constructed on a 90ST quartz sub-strate, the

( )

^{1 1 and}

^{( )}

¹¹ rows of the square lattice are no longer lines of mirror symmetry while rows

( )

^{10 and}

( )

⁰¹ still remain, therefore the irreducible Brillouin zone should be the square surrounded by the four points Γ, X, M, and Y.

2.2.2 Bloch’s theorem

The special form of the solutions of differential equations with periodic coeffi-cients is given by Bloch’s theorem. It dates back to 1883, Floquet [45] first showed that the solutions of a 1-variable n × n system of linear differential equations with the co-efficients being a piecewise continuous periodic function with period a

where ηηk is a periodic function with period a. This result can be extended and applied to higher order differential equations by reducing the equation(s) to first order equations with increasing the dimension of the system. It is known as Floquet’s theorem. And in

1929, Bloch generalized the theorem for three dimensions in his work on the electron in a crystal [46]. We adopt his result and we have that the eigenmodes of a wave equation in a periodic medium (1-D, 2-D, or 3-D) may be written as the product of a plane wave function _e^ik⋅x and a periodic function _{ηη x}

( )

that has the same periodicity as the

where T is the translation vector of the crystal that is a linear combination of the basis vectors of the direct lattice with integer coefficients. We can also define G as the trans-lation vector in the reciprocal lattice.

Applying the Bloch’s theorem we obtain the boundary conditions of a Wig-ner-Seitz cell. Then we search the eigenmodes with wave vectors k’s located in the ir-reducible Brillouin zone, than the Bloch waves can be completely characterized. We make the dispersion relation (the ω-k relation, where ω is the eigenfrequency) plotted with k varying along the boundary of the irreducible Brillouin zone, which is call the

band structure, adopted from the terminology of solid state physics. The frequency

bands that contain no propagating modes in the band structure are called the band gaps, in which no wave can propagate in the periodic structure.

2.3 Love wave dispersion of a layered half-space

In this section we conduct the calculation of the acoustic dispersion relation of the layered structure comprises a silica film deposited on a 90ST quartz substrate. It was accomplished by applying the Bloch’s theorem and the finite element method (FEM).

And a proper thickness of the silica film is proposed for the experiment devices.

2.3.1 Numerical methods

We are considering the Love wave propagating on the layered structure that is ho-mogeneous along the x-y plane. But if we treat it as a periodic structure with “vanish-ingly small variation,” the Bloch’s theorem can be applied to calculate the dispersions.

Since the structure is actually homogeneous, the choice of the unit cell size can be whatever we like. Let the cell have a length a on the x-direction (i.e., have a “period” a), and a piezoelectric acoustic wave with wavenumber k is propagating through this 1-D

“lattice,” then the Bloch wave function can be represented as

Where here _ψ^ψ_k can be the displacement field, velocity field, strain field, stress field, electric potential, electric displacement, etc. Therefore we can write down the boundary conditions of this cell with the dependent variable u and φ

where the subscription + and – stand for the properties on the right and left boundaries of the cell respectively. This is sometimes called Floquet periodicity. Generally a con-tinuous condition requires constraints on both the displacement and traction (electric potential and electric displacement) on the identity pair boundaries. But since we know ηη^k(x) in equation (2.29) for u and φ are smooth functions, we set the constraint order of boundary conditions (2.31) to be 2 as the quadratic Lagrangian element is used in the FEM. Therefore the traction and electric displacement also satisfying condition (2.29)

for they are just results of linear operation on u and φ. Thus they are automatically con-strained similar to boundary conditions (2.31). And with the governing equations of the domain (2.14), (2.15), the dispersion relation can be obtain by finding the eigenfre-quency ω with varying the wavenumber k.

It is expected that all the dispersion information _{k ∈ 0, ∞}

( )

is completely

deter-mined in the irreducible Brillouin zone that is a finite interval k ∈ 0, π a⎡⎣ ⎤⎦. As a result we see the dispersion curves fold back when they touch the boundary of the irreducible Brillouin zone. It is a natural result for e^ika is a periodic function of k with period 2π/a, and there is no difference in velocity between the right- or left-propagating waves. So we have a dispersion that is periodic and even function of wavenumber k. We may also consider the boundary conditions (2.31) as “wave propagating boundary conditions”

that a wave with wavenumber k propagating through a medium with a length a, experi-ences a phase shift ka, so the wave function on the right boundary is that on the left boundary being multiplied by e^ika. Consider propagating waves with wavenumber

(

^{2 1}

)

all identical to (2.31) for the periodicity and of the exponential function.

We would like to postpone the folding on the boundary of the irreducible Brillouin zone to make the dispersion curve look simpler. It can be achieved by selecting a smaller unit cell, therefore the irreducible Brillouin zone, which is a unit cell in the re-ciprocal lattice, becomes larger, and the folding is postponed.

Ideally the model is a layered half-space, that is, the substrate is infinite deep along –z-direction. However we cannot build a model with infinite depth in FEM calcu-lation. Also the perfect matching layer (PML), and some absorption or low reflecting boundaries are not suitable in the eigenfrequency analysis. Therefore instead, a

fi-nite-depth model is used with the bottom boundary set to be fixed (u = 0) and zero charged (Dz = 0). As a result, we are actually calculating the dispersion of a “plate.” But the SAW behavior can be found on the plate as the wavelength becomes much smaller than the plate thickness, or

kh

_substrate >> [47] Thus, if the thickness of the model’s 1.

substrate is thick enough, we can still obtain the SAW dispersion with such model. Be-ing a surface wave, the Love wave has a displacement field in the substrate exponen-tially vanishing along the –z-direction as

[40] We can see the vanishing factor b depends on both wave number k and the phase velocity of the Love wave cLove. And the phase velocity cLove always sets off from the substrate’s shear wave velocity cSH, 90ST, and approaches to the shear wave velocity of the film material cS, silica as k increasing. A thickness-varying (with k) model and a fixed thickness model are employed to calculate the dispersion of the Love wave, and they show similar results (Fig 2.6) after mode selection (described in the next paragraph).

Considering the penetration depth of Love waves is not only wavelength dependent (check “b” in equation (2.32)), and the thickness-varying model requires huge compu-tation in the small k region which we do not really concern, so henceforth fixed thick-ness models with sufficient thickthick-ness are used instead of the thickthick-ness-varying models.

To pick out the modes standing for Love waves among numerous plate modes in the dispersion relation calculated by the finite-depth model, two conditions are set for mode selecting. We first define the depth of energy (DOE) as

which is the depth of the strain-energy-weighted centroid of the unit cell. The star sym-bol “*” indicates the complex conjugate value of the calculated mode shape. Since the stress and strain are always in phase, the amplitude of the strain energy can be expressed

as ^{1 1 *}.

2T S^{ij ij} =2T S^{ij ij} By setting a threshold of DOE (for example 0.2λ), the SAW modes whose energy concentrate near the free surface filters through. The selected SAW modes contain generalized Rayleigh waves (particle motion parallel to the sagittal plane), Love waves (particle motion perpendicular to the sagittal plane), and modes lie somewhere in between. We make a threshold of polarization to execute a dichotomy between them. First the displacement component normal to the sagittal plane (i.e. the shear horizontal component) of a wave with the propagation vector lies on the xy-plane

2T S^{ij ij} =2T S^{ij ij} By setting a threshold of DOE (for example 0.2λ), the SAW modes whose energy concentrate near the free surface filters through. The selected SAW modes contain generalized Rayleigh waves (particle motion parallel to the sagittal plane), Love waves (particle motion perpendicular to the sagittal plane), and modes lie somewhere in between. We make a threshold of polarization to execute a dichotomy between them. First the displacement component normal to the sagittal plane (i.e. the shear horizontal component) of a wave with the propagation vector lies on the xy-plane

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