**2. Acoustic modes propagated in piezoelectric layered structures**

**1. Introduction**

162 Piezoelectric Materials

up to 20 GHz [3].

Known piezoelectric materials such as quartz, lithium niobate, lithium tantalate, langasite, etc. have been widely used as the substrates in a number of acoustoelectronic devices such as resonators, acoustic filters, delay lines, sensors, etc. Such devices can operate on one or several typesofacousticwaves,suchasconventionalbulk(BAW)acousticwaves,surface(SAW)acoustic waves (Rayleigh, Sezawa, Gulyaev-Bleustein, shear-horizontal (*SH*), and Love waves), normal plate waves of Lamb-type, and Stoneley acoustic waves on the interface between two differ‐ ent solids. In the twenty-first century, state-of-art technologies have a clear requirement for passive electronic and acoustoelectronic components operating at high and ultra-high (UHF) frequencies with low insertion losses. Previously, conventional structures and materials could not be applied at UHF due to their high acoustic attenuation. Possible solution is to use piezoelectric layered structures (PLS) based on appropriate single crystalline substrates with low UHF acoustic attenuation. Modern precise technologies of thin-film deposition provide fabrication of submicron piezoelectric layers with excellent parameters. Hexagonal alumi‐ numnitride(AlN)filmsareusedmoreoftenthanzincoxide(ZnO)filmsduetotheirbestdielectric properties and thermal stability. Applying PLS approach, one can extend a set of substrate materials to be used up to non-piezoelectric crystals with outstanding physical properties.

Well-known types of BAW resonators are the conventional piezoelectric resonators more often produced out of crystalline quartz, including the high-frequency resonator as inverse mesa structure, thin-film bulk acoustic resonators (FBAR) [1], and solidly mounted resonators (SMR) [1,2], but they have significantly lower both *Q* factor and operating frequencies compared to high overtone bulk acoustic resonator (HBAR). Such HBARs can be used at high frequencies up to 10 GHz because the elastic energy is mainly concentrated within the substrate material [3]. Usually HBARs are designed as PLS with a specified electrode structure deposed on a crystalline substrate with low UHF acoustic attenuation. In earlier investigated BAW resona‐ tors, single crystalline and fused quartz, silicon [4], sapphire [4,5], and yttrium aluminum garnet (YAG) substrates [6] were used. It was proved that diamond single crystal has a low acoustic attenuation at UHF and diamond-based HBARs are known operating at frequencies

The choice of HBAR's substrate material is a problem of high importance since it is necessary to take into account an appropriate combination of physical and chemical properties, low acoustic attenuation at UHF, crystalline quality, possibility of precise treatment, etc. It is well known that physical properties of thin films such as the density and elastic constants can considerably differ from the ones measured on the bulk specimens. HBAR can be used as an instrument for determination of material properties of substrates and thin films at microwave frequencies with high accuracy. Additionally, HBAR application gives a unique possibility to measure frequency-sensitive properties, for example, the acoustic attenuation of substrate's

Application of diamond as a substrate material in this chapter is caused by its unique physical and acoustical properties: it is the hardest crystal with highest BAW and SAW velocities (in

= 18860 m/s [7]); it has high

material within a wide frequency range of 0.5–10 GHz.

[111] direction, the phase velocity of BAW longitudinal type *vl*

#### **2.1. Piezoelectric layered structure as a complex acoustic system: normal bulk, surface and plate acoustic waves**

Propagation of the small amplitude acoustic waves in the piezoelectric crystal is described by the equations of motion and electrostatics and, additionally, the equations of state of the piezoelectric medium as [11]:

$$\begin{aligned} \mathfrak{p}\_0 \ddot{\tilde{U}}\_l &= \mathfrak{\tilde{\mathfrak{r}}}\_{\,\,\,k}, \\ \tilde{D}\_{\,\,m,n} &= 0, \\ \tilde{\mathfrak{\tau}}\_{\,\,kl} &= C\_{\,\,kpq}^E \tilde{\mathfrak{\eta}}\_{\,\,pq} - e\_{nk} \tilde{E}\_n, \\ \tilde{D}\_n &= e\_{nk} \tilde{\mathfrak{\eta}}\_{\,\,kl} + \varepsilon\_{mn}^{\eta} \tilde{E}\_n. \end{aligned} \tag{1}$$

There were introduced the values to be used as follows: ρ<sup>0</sup> is the crystal density; *U*˜ *<sup>i</sup>* is the unit vector of the dynamical elastic displacement; *τik* is the thermodynamical stress tensor; *ηab* is the strain tensor; *Em* and *Dm* are the vectors of electric field and electric displacement, respectively; *Cikpq <sup>E</sup>* , *enik*, and *εmn* <sup>η</sup> are the elastic, piezoelectric, and clamped dielectric constants, respectively. Here and after, the time-dependent variables are marked by "tilde" symbol. The comma after the subscript denotes that the spatial derivatives and coordinate Latin indices vary from 1 to 3. Here and further, the rule of summation over repeated indices is used.

Taking into account the dispersion of the elastic waves, a general form of the Christoffel equation and its components can be written as follows:

$$\begin{aligned} \left(\Gamma\_{ik} - \rho\_0 \alpha^2 \delta\_{ik}\right) \mathfrak{a}\_i = 0; i, k = 1, \ldots, 4; \mathfrak{d}\_{44} = 0;\\ \Gamma\_{ik} = C\_{ijkm}^E k\_j k\_m; \\ \Gamma\_{4\_f} = \Gamma\_{\ne 4} = e\_{jk} k\_i k\_j, \Gamma\_{44} = -\varepsilon\_{nm}^\eta k\_n k\_m; \end{aligned} \tag{2}$$

where *k* <sup>→</sup> =(*ω* / *v*)*n* <sup>→</sup> is the wave vector and *n* <sup>→</sup> is the unit vector of the wave normal; α*<sup>i</sup>* are the eigenvectors (components of wave elastic displacement); α4 is the amplitude of the wave of quasi-static electrical potential; ω is the circular frequency, and υ is the phase velocity. Solution of Christoffel equation as a standard problem of the eigenvalues and eigenvectors brings determining the characteristics of the bulk acoustic wave in a piezoelectric crystal.

The propagation of elastic waves in the multilayer piezoelectric structure should be written introducing additionally the boundary conditions depending on the number *m* of layer:

$$\begin{aligned} \left. \pi\_{i}^{(1)} = \mathbf{0} \right|\_{s\_{1} = \boldsymbol{b}\_{1}}, \qquad & D\_{3}^{(1)} = D^{(\max)} \Big|\_{s\_{1} = \boldsymbol{b}\_{1}}, \\ \left. \pi\_{i}^{(1)} = \pi\_{j}^{(2)} \right|\_{s\_{1} = \boldsymbol{b}\_{2}}, \qquad & D\_{3}^{(1)} = D\_{3}^{(2)} \Big|\_{s\_{1} = \boldsymbol{b}\_{2}}, \\ \left. \Phi^{(1)} = \mathbf{0}^{(2)} \right|\_{s\_{1} = \boldsymbol{b}\_{2}}, \qquad & U\_{i}^{(1)} = U\_{i}^{(2)} \Big|\_{s\_{1} = \boldsymbol{b}\_{2}}, \\ & \qquad & \cdots \\ \left. \pi\_{j}^{(m-1)} = \pi\_{j}^{(m)} \right|\_{s\_{1} = 0}, \qquad & D\_{3}^{(m-1)} = D^{(m)} \Big|\_{s\_{1} = 0}, \\ \left. \Phi^{(m-1)} = \boldsymbol{\Phi}^{(m)} \right|\_{s\_{1} = 0}, \qquad & U\_{i}^{(m-1)} = U\_{i}^{(m)} \Big|\_{s\_{1} = 0}, \\ \left. \pi\_{j}^{(m)} = \boldsymbol{0} \right|\_{s\_{1} = \boldsymbol{b}\_{2}}, \qquad & D\_{3}^{(m)} = D^{(m)} \Big|\_{s\_{1} = \boldsymbol{b}\_{2}}. \end{aligned} \tag{3}$$

Here *hm* is the thickness of the *m-*th layer, and *X*3 axis coincides with the vertical direction. Substituting the elastic displacements and electric potential taken in the form of linear combinations of *n* partial waves into the boundary conditions (3) as

$$\begin{aligned} U\_i^{(n)} &= \sum\_{\mathbf{x}} C\_n^{(n)} \mathbf{a}\_i^{(n)} \exp \left[ i \left( k\_1 \mathbf{x}\_1 + k\_3^{(n)} \mathbf{x}\_3 - \alpha t \right) \right], \\ \Phi^{(n)} &= \sum\_{\mathbf{x}} C\_4^{(n)} \mathbf{a}\_4^{(n)} \exp \left[ i \left( k\_1 \mathbf{x}\_1 + k\_3^{(s)} \mathbf{x}\_3 - \alpha t \right) \right], \end{aligned} \tag{4}$$

one can obtain a matrix of the boundary conditions. Here, *Cn* (*m*) and *C*<sup>4</sup> (*m*) are the amplitude coefficients of elastic displacements and electric potential in the *m-*th layer, respectively. Equating the matrix determinant to zero forms an algebraic equation determining the charac‐ teristics of the elastic wave. The variations of the boundary conditions (3) will define all the types of elastic waves propagating in a layered structure. For example, the first equation in Eq (3) determines propagation of Rayleigh-type waves, and the first and last equations taken at *m* = 1 describe the propagation of elastic waves in a piezoelectric plate,that is, now Eq (3) can be written as

$$\begin{cases} \sum\_{s=1}^{s} \left( C\_{\,j,al} k\_{l}^{(s)} \mathbf{a}\_{k}^{(s)} + e\_{k\,j} k\_{k}^{(s)} \mathbf{a}\_{4}^{(s)} \right) \exp\left( i k\_{3}^{(s)} h \right) = \mathbf{0}, \\\sum\_{s=1}^{s} C\_{\,s} \left[ e\_{\,j,l} k\_{l}^{(s)} \mathbf{a}\_{k}^{(s)} - \left( \varepsilon\_{3i}^{n} k\_{k}^{(s)} - i \varepsilon\_{0} \right) \mathbf{a}\_{4}^{(s)} \right] \exp\left( i k\_{3}^{(s)} h \right) = \mathbf{0}, \\\sum\_{s=1}^{s} C\_{\,s} \left( C\_{\,j,al} k\_{l}^{(s)} \mathbf{a}\_{k}^{(s)} + e\_{k\,j,l} k\_{k}^{(s)} \mathbf{a}\_{4}^{(s)} \right) = \mathbf{0}, \\\sum\_{s=1}^{s} C\_{\,s} \left[ e\_{\,3d} k\_{l}^{(s)} \mathbf{a}\_{k}^{(s)} - \left( \varepsilon\_{3i}^{n} k\_{k}^{(s)} + i \varepsilon\_{0} \right) \mathbf{a}\_{4}^{(s)} \right] = \mathbf{0}. \end{cases} \tag{5}$$

If we assume that the lower layer is thick enough (semi-infinite type), that is, its thickness is much greater than the length of the elastic wave, then in this case the latter equation in Eq. (3) can be ignored, that is, a free bottom border takes place. It is also necessary to require the fulfillment of the condition which provides attenuation of elastic waves in the substrate as *Im*(*k*<sup>3</sup> (*m*) ) <0. Then the equations describing the propagation of elastic waves in the "layer– substrate" structure will look like

quasi-static electrical potential; ω is the circular frequency, and υ is the phase velocity. Solution of Christoffel equation as a standard problem of the eigenvalues and eigenvectors brings

The propagation of elastic waves in the multilayer piezoelectric structure should be written introducing additionally the boundary conditions depending on the number *m* of layer:

3 1 3 1

= =

*D D*

3 2 3 2

= =

*D D*

3 2 3 2

= =

*U U*

*i i x h x h*

3 3

3 3

*D D*

=- =-

Here *hm* is the thickness of the *m-*th layer, and *X*3 axis coincides with the vertical direction. Substituting the elastic displacements and electric potential taken in the form of linear

11 3 3

<sup>å</sup> (4)

(*m*)

 and *C*<sup>4</sup> (*m*)

are the amplitude

α exp ω ,

*U U*

= =

*i i x x*

0 0

, ;

(3)

*D D*

= =

*vac*

<sup>3</sup> <sup>3</sup>

τ0 , . *<sup>m</sup> <sup>m</sup>*

( ) ( ) ( ) ( ) ( )

*U C i kx k x t*

*m mn n*

*m mn n*

( ) ( ) ( ) ( ) ( )

Φ α exp ω ,

4 4 11 3 3

<sup>=</sup> é ù + - ë û

<sup>=</sup> é ù + - ë û

*C i kx k x t*

coefficients of elastic displacements and electric potential in the *m-*th layer, respectively. Equating the matrix determinant to zero forms an algebraic equation determining the charac‐ teristics of the elastic wave. The variations of the boundary conditions (3) will define all the types of elastic waves propagating in a layered structure. For example, the first equation in Eq (3) determines propagation of Rayleigh-type waves, and the first and last equations taken at *m* = 1 describe the propagation of elastic waves in a piezoelectric plate,that is, now Eq (3)

determining the characteristics of the bulk acoustic wave in a piezoelectric crystal.

( ) ( ) ( )

= =

τ 0 , ;

*<sup>j</sup> x h x h*

ττ , ;

*j j x h x h*

φφ , ;

1 1 3 3

( ) ( ) ( ) ( )

12 1 2 33 3 3 12 1 2

= =

( ) ( ) ( ) ( )

= =

1 1 33 3 0 0


τ τ

1 1


3 3

combinations of *n* partial waves into the boundary conditions (3) as

*i ni n*

å

*n*

one can obtain a matrix of the boundary conditions. Here, *Cn*

can be written as

164 Piezoelectric Materials

( ) ( ) ( ) ( )

= =

*mm m m j j x x mm m m*

...

( ) ( ) ( ) ( )

= =

φφ, ;

( ) ( ) ( )

= =

*m m vac <sup>j</sup> x h x h*

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) <sup>8</sup> 1 1 3 43 4 3 1 <sup>8</sup> 1 1 3 43 0 4 3 1 4 8 22 1 1 3 34 3 34 1 1 4 8 2 2 3 34 3 1 1 α α exp 0, α ε ε α exp 0, α α α α 0, αε α = = = = = = + = é ù +- = ë û +- + = + å å å å å å *n n nn n n kpi i p p k p n n n nn n n kl l k p p n ss ss nn nn m i kl l k pi p n i kl l k pi p m n ss ss m kl l k p p n k m n a C k a e k ik h ae k a k i ik h bC k ek aC k ek b ek k ae*( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) 1 1 3 4 4 8 2 1 1 1 α ε α 0, 0. = = + = å å- = *nn nn ll k p p s n im in s n k k Ub Ua* (6)

Digital superscripts 1 and 2 denote the layer and the substrate, respectively; α*<sup>k</sup>* (*s*) and *bm* are the magnitudes and weight coefficients of *s*-th partial wave (*s* = 1, …, 4 ) in the substrate; α*<sup>k</sup>* (*n*) and *an* are the magnitudes and weight coefficients of *n*-th partial wave (*n* = 1, …, 8 ) in the piezo‐ electric layer.

#### **2.2. Dispersive relations for the elastic waves in plates and layered structures**

Before investigation of the propagation of elastic waves in a multilayer structure such as "Me1/ AlN/Me2/diamond," it makes sense initially to study the marginal cases as propagation of elastic waves in a plate and a layered structure (interlayer interface).

Let us perform the analysis of the characteristics of an acoustic wave in piezoelectric crystalline plate belonging to the point symmetry 23. All the results obtained will be reasonable for other cubic crystals too. In this case, Christoffel tensor Eq. (2) written for the propagation of elastic wave along the [100] direction in (001) plane takes the following form:

$$
\Gamma\_{\boldsymbol{d}} = \begin{bmatrix}
\boldsymbol{C}\_{11}\boldsymbol{k}\_{1}^{2} + \boldsymbol{C}\_{44}^{E}\boldsymbol{k}\_{3}^{2} & \boldsymbol{0} & \left(\boldsymbol{C}\_{11} + \boldsymbol{C}\_{44}^{E}\right)\boldsymbol{k}\_{1}\boldsymbol{k}\_{3} & \boldsymbol{0} \\
\boldsymbol{0} & \boldsymbol{C}\_{44}^{E}\left(\boldsymbol{k}\_{1}^{2} + \boldsymbol{k}\_{3}^{2}\right) & \boldsymbol{0} & 2\boldsymbol{e}\_{14}\boldsymbol{k}\_{1}\boldsymbol{k}\_{3} \\
\left(\boldsymbol{C}\_{11} + \boldsymbol{C}\_{44}^{E}\right)\boldsymbol{k}\_{1}\boldsymbol{k}\_{3} & \boldsymbol{0} & \boldsymbol{C}\_{44}^{E}\boldsymbol{k}\_{1}^{2} + \boldsymbol{C}\_{11}\boldsymbol{k}\_{3}^{2} & \boldsymbol{0} \\
\boldsymbol{0} & 2\boldsymbol{e}\_{14}\boldsymbol{k}\_{1}\boldsymbol{k}\_{3} & \boldsymbol{0} & -\boldsymbol{e}\_{11}^{\eta}\left(\boldsymbol{k}\_{1}^{2} + \boldsymbol{k}\_{3}^{2}\right) \\
\end{bmatrix}.
\tag{7}$$

Obviously, in this case Γ*il* tensor can be decomposed into two independent parts, correspond‐ ing to the partial wave motion in the *X*1*X*<sup>3</sup> sagittal plane and along the *X*2 transverse direction. As usual, the *X*1 axis of special Cartesian coordinate system coincides with the wave propa‐ gation direction. First part describes a nonpiezoactive Lamb wave, and the second one corresponds to the *SH* acoustic wave possessing the piezoelectric activity.

Characteristic Eq. (7) with respect to *k*<sup>3</sup> for the Lamb-type wave is a polynomial of the fourth power:

$$\left(C\_{11}k\_1^2 + C\_{44}^E k\_3^2 - \mathfrak{p}\_0 \alpha^2\right) \left(C\_{44}^E k\_1^2 + C\_{11} k\_3^2 - \mathfrak{p}\_0 \alpha^2\right) - \left(C\_{12} + C\_{44}^E\right)^2 k\_1^2 k\_3^2 = 0\tag{8}$$

Roots of Eq. (8) take the following solutions:

$$k\_{\circ}^{(n)} = \pm i \sqrt{\frac{1}{2R} \left( P \mp \sqrt{P^2 - 4RQ} \right)},\tag{9}$$

where *R* =*C*11*C*<sup>44</sup> *<sup>E</sup>*, *<sup>P</sup>* <sup>=</sup>*C*11(*C*11*<sup>k</sup>* <sup>2</sup> −ρ0ω<sup>2</sup> ) + *C*<sup>44</sup> *<sup>E</sup>*(*C*<sup>44</sup> *Ek* <sup>2</sup> −ρ0ω<sup>2</sup> ) −(*C*<sup>11</sup> + *C*<sup>44</sup> *<sup>E</sup>*)<sup>2</sup> *k* 2 , *<sup>Q</sup>* =(*C*11*<sup>k</sup>* <sup>2</sup> - <sup>ρ</sup>0ω<sup>2</sup> )(*C*<sup>44</sup> *Ek* <sup>2</sup> - <sup>ρ</sup>0ω<sup>2</sup> ), and *k*≡*k*1. Eigenvectors corresponding to *k*<sup>3</sup> (*n*) values can be written as

$$\begin{aligned} k\_3^{(1)} &= q\_{1'}, & \alpha^{(1)} &= \{1, 0, p\_1, 0\}; \\ k\_3^{(2)} &= -q\_1 = -k\_3^{(1)}, & \alpha^{(2)} &= \{1, 0, -p\_1, 0\}; \\ k\_3^{(3)} &= q\_3, & \alpha^{(3)} &= \{1, 0, p\_3, 0\}; \\ k\_3^{(4)} &= -q\_3 = -k\_3^{(3)}, & \alpha^{(4)} &= \{1, 0, -p\_3, 0\}; \end{aligned} \tag{10}$$

$$\text{where } p\_1 = -\frac{\mathbb{C}\_{11}\mathbb{k}^2 - \mathbb{q}\_0\omega^2 + \mathbb{C}\_{44}^\mathbb{k}\eta^2}{\{\mathbb{C}\_{12} + \mathbb{C}\_{44}^\mathbb{k}\}\mathbb{k}\eta\_1} \text{ and } p\_3 = -\frac{\mathbb{C}\_{11}\mathbb{k}^2 - \mathbb{q}\_0\omega^2 + \mathbb{C}\_{44}^\mathbb{k}\eta\_3^2}{\{\mathbb{C}\_{12} + \mathbb{C}\_{44}^\mathbb{k}\}\mathbb{k}\eta\_3}.$$

Taking Eq. (10) into account, one can obtain the system of Eq. (5) with respect to components α(*n*) and *k*(*n*) , and then the determinant can be defined as follows:

$$\begin{bmatrix} Ae^{iq\_hh} & -Ae^{-iq\_fh} & Be^{iq\_hh} & -Be^{-iq\_fh} \\ Ce^{iq\_hh} & Ce^{-iq\_hh} & De^{iq\_hh} & De^{-iq\_hh} \\ & A & -A & B & -B \\ C & C & & D & D \end{bmatrix},\tag{11}$$

where

Let us perform the analysis of the characteristics of an acoustic wave in piezoelectric crystalline plate belonging to the point symmetry 23. All the results obtained will be reasonable for other cubic crystals too. In this case, Christoffel tensor Eq. (2) written for the propagation of elastic

( )

0 0

44 1 3 14 1 3 2 2

*Ck k e kk*

0 0

11 1 3 14 1 3

*e kk k k*

11 1 44 3 0 44 1 11 3 0 12 44 1 3 ρ ω ρ ω 0 *E E <sup>E</sup> Ck C k C k Ck C C kk* + - + - -+ = (8)

*<sup>n</sup> k i P P RQ <sup>R</sup>* <sup>=</sup> ± - <sup>m</sup> (9)

*<sup>E</sup>*)<sup>2</sup> *k* 2 ,

(*n*)

values can be

(10)

) −(*C*<sup>11</sup> + *C*<sup>44</sup>

( )

h

e

2 2

(7)

wave along the [100] direction in (001) plane takes the following form:

*E*

*il <sup>E</sup> <sup>E</sup>*

( )

Roots of Eq. (8) take the following solutions:

3

*<sup>E</sup>*, *<sup>P</sup>* <sup>=</sup>*C*11(*C*11*<sup>k</sup>* <sup>2</sup> −ρ0ω<sup>2</sup>

*Ek* <sup>2</sup> - <sup>ρ</sup>0ω<sup>2</sup>

)(*C*<sup>44</sup>

power:

166 Piezoelectric Materials

where *R* =*C*11*C*<sup>44</sup>

*<sup>Q</sup>* =(*C*11*<sup>k</sup>* <sup>2</sup> - <sup>ρ</sup>0ω<sup>2</sup>

written as

2 2

<sup>+</sup> <sup>=</sup>

( )

*Ck Ck C C kk*

11 44 1 3 44 1 11 3

*C C kk Ck Ck*

+ +

corresponds to the *SH* acoustic wave possessing the piezoelectric activity.

( )( ) ( )

) + *C*<sup>44</sup> *<sup>E</sup>*(*C*<sup>44</sup>

<sup>2</sup> 2 22 2 22 2 2

( ) ( ) <sup>2</sup>

<sup>1</sup> 4 , <sup>2</sup>

*Ek* <sup>2</sup> −ρ0ω<sup>2</sup>

), and *k*≡*k*1. Eigenvectors corresponding to *k*<sup>3</sup>

0 2 0

11 1 44 3 11 44 1 3 2 2

*E E*

0 0 2 Γ .

é ù <sup>+</sup> <sup>+</sup> ê ú


Obviously, in this case Γ*il* tensor can be decomposed into two independent parts, correspond‐ ing to the partial wave motion in the *X*1*X*<sup>3</sup> sagittal plane and along the *X*2 transverse direction. As usual, the *X*1 axis of special Cartesian coordinate system coincides with the wave propa‐ gation direction. First part describes a nonpiezoactive Lamb wave, and the second one

Characteristic Eq. (7) with respect to *k*<sup>3</sup> for the Lamb-type wave is a polynomial of the fourth

$$\begin{aligned} p\_1 &= -\frac{C\_{11}k^2 + C\_{44}^E q\_1^2 - p\_0 \mathbf{o}^2}{\left(C\_{12} + C\_{44}^E\right) k q\_1}, \\ p\_3 &= -\frac{C\_{11}k^2 + C\_{44}^E q\_3^2 - p\_0 \mathbf{o}^2}{\left(C\_{12} + C\_{44}^E\right) k q\_3}, \\ A &= C\_{44}^E \left(k p\_1 + q\_1\right); B = C\_{44}^E \left(k p\_3 + q\_3\right), \\ C &= C\_{12}k + C\_{11}q\_1 p\_1; D = C\_{12}k + C\_{11}q\_3 p\_3. \end{aligned} \tag{12}$$

Equating to zero the determinant of the matrix (11), one can get the equation describing the propagation for the symmetric mode:

$$\frac{\text{th}\left(\frac{1}{2}iq\_{\text{j}}\hbar\right)}{\text{th}\left(\frac{1}{2}iq\_{\text{j}}\hbar\right)} = \frac{q\_{\text{3}}}{q\_{\text{1}}} \frac{\left[C\_{11}\left(k^{2}-k\_{\text{j}}^{2}\right) - C\_{12}q\_{\text{1}}^{2}\right]\left[C\_{12}\left(C\_{12}+C\_{44}^{E}\right)k^{2} - C\_{11}\left[C\_{11}\left(k^{2}-k\_{\text{j}}^{2}\right) + C\_{44}^{E}q\_{\text{3}}^{2}\right]\right]}{q\_{\text{1}}\left[C\_{11}\left(k^{2}-k\_{\text{j}}^{2}\right) - C\_{12}q\_{\text{3}}^{2}\right]\left[C\_{12}\left(C\_{12}+C\_{44}^{E}\right)k^{2} - C\_{11}\left[C\_{11}\left(k^{2}-k\_{\text{j}}^{2}\right) + C\_{44}^{E}q\_{\text{1}}^{2}\right]\right]}\tag{13}$$

and for the antisymmetric mode of Lamb wave [12]:

$$\frac{\text{ch}\left(\frac{1}{2}iq\_{3}h\right)}{\text{ch}\left(\frac{1}{2}iq\_{1}h\right)} = \frac{q\_{1}}{q\_{3}}\frac{\left[C\_{11}\left(k^{2}-k\_{l}^{2}\right)-C\_{12}q\_{3}^{2}\right]\left[C\_{12}\left(C\_{12}+C\_{44}^{E}\right)k^{2}-C\_{11}\left[C\_{11}^{E}\left(k^{2}-k\_{l}^{2}\right)+C\_{44}^{E}q\_{1}^{2}\right]\right]}{\left[C\_{11}\left(k^{2}-k\_{l}^{2}\right)-C\_{12}q\_{1}^{2}\right]\left[C\_{12}\left(C\_{12}+C\_{44}^{E}\right)k^{2}-C\_{11}\left[C\_{11}^{E}\left(k^{2}-k\_{l}^{2}\right)+C\_{44}^{E}q\_{3}^{2}\right]\right]}.\tag{14}$$

Here *kl* =( ω *vl* ) is modulus of the wave vector of the longitudinal bulk acoustic wave, and *qn*≡*k*<sup>3</sup> (*n*) is the solution of biquadratic dispersive Eq. (8).

For *SH*-waves, the characteristic dispersive equation similar to Eq. (8) can be written as

$$-\mathfrak{e}\_{11}^{\eta} \left[ C\_{44}^{E} \left( k\_1^2 + k\_3^2 \right) - \rho\_0 \alpha \alpha^2 \right] \left( k\_1^2 + k\_3^2 \right) - 4 \mathfrak{e}\_{14}^2 k\_1^2 k\_3^2 = 0. \tag{15}$$

Solving Eq. (15) with respect to *k*3 at the parameter *k*≡*k*1, one can obtain the result:

$$k\_{\rm 3}^{(1,2,3,4)} = \pm i \sqrt{k^2 - \frac{1}{2} \left[ k\_{\rm \prime}^2 - 4k\_{\rm 14}^2 k^2 \pm \sqrt{\left( k\_{\rm \prime}^2 - 4k\_{\rm 14}^2 k^2 \right)^2 + 16k\_{\rm 14}^2 k^4} \right]}.\tag{16}$$

Here *kt* =( ω *vt* ) is the modulus of wave vector of the shear bulk acoustic wave, and *k*<sup>14</sup> <sup>=</sup> *<sup>e</sup>*<sup>14</sup> ε11 <sup>η</sup> *<sup>C</sup>*<sup>44</sup> *E* is the electromechanical coupling coefficient (EMCC). Eigenvectors corresponding to a value *k*3 (*n*) are equal to

$$\begin{aligned} k\_3^{(1)} &= q\_{2'}, & \alpha^{(1)} &= \left(0, 1, 0, p\_{\pi}\right); \\ k\_3^{(2)} &= -k\_3^{(1)} = -q\_{2'}, & \alpha^{(2)} &= \left(0, 1, 0, -p\_{\pi}\right); \\ k\_3^{(3)} &= q\_{4'}, & \alpha^{(3)} &= \left(0, 1, 0, p\_{\pi}\right); \\ k\_3^{(4)} &= -k\_3^{(3)} = -q\_4, & \alpha^{(4)} &= \left(0, 1, 0, -p\_4\right). \end{aligned} \tag{17}$$

where *p*<sup>2</sup> =α<sup>4</sup> (1) <sup>=</sup> <sup>2</sup>*e*14*<sup>k</sup> <sup>q</sup>*<sup>2</sup> ε11 <sup>η</sup> (*<sup>k</sup>* <sup>2</sup> <sup>+</sup> *<sup>q</sup>*<sup>2</sup> 2) , *p*<sup>4</sup> =α<sup>4</sup> (3) <sup>=</sup> <sup>2</sup>*e*14*<sup>k</sup> <sup>q</sup>*<sup>4</sup> ε11 <sup>η</sup> (*<sup>k</sup>* <sup>2</sup> <sup>+</sup> *<sup>q</sup>*<sup>4</sup> 2) ..

As a result, in this case, the determinant of the boundary conditions (Eq. (5)) is obtained as follows:

$$
\begin{bmatrix}
A e^{iq\_2h} & -A e^{-iq\_2h} & B e^{iq\_4h} & -B e^{-iq\_4h} \\
(C+D)e^{iq\_2h} & (C-D)e^{-iq\_2h} & (E+F)e^{iq\_4h} & (E-F)e^{-iq\_4h} \\
A & -A & B & -B \\
(C-D) & (C+D) & (E-F) & (E+F)
\end{bmatrix},\tag{18}
$$

where

$$\begin{aligned} A &= \mathbb{C}\_{44}^{\mathbb{E}} q\_2 + e\_{14} k p\_{2 \prime} & B = \mathbb{C}\_{44}^{\mathbb{E}} q\_4 + e\_{14} k p\_{4 \prime} & \mathbb{C} &= e\_{14} k - \mathbb{c}\_{11}^{\eta} q\_2 p\_{2 \prime} \\ D &= i \varepsilon\_0 p\_{2 \prime} & E &= e\_{14} k - \mathbb{c}\_{11}^{\eta} q\_4 p\_{4 \prime} \\ \end{aligned} \tag{19}$$

Taking the similar procedure as for the Lamb wave solution, the matrix of the boundary conditions (18) is divided into two independent parts, and the dispersive equations describing the propagation of *SH*-waves could be obtained:

Here *kl* =(

168 Piezoelectric Materials

Here *kt* =(

where *p*<sup>2</sup> =α<sup>4</sup>

follows:

where

(1)

<sup>=</sup> <sup>2</sup>*e*14*<sup>k</sup> <sup>q</sup>*<sup>2</sup> ε11 <sup>η</sup> (*<sup>k</sup>* <sup>2</sup> <sup>+</sup> *<sup>q</sup>*<sup>2</sup>

2) , *p*<sup>4</sup> =α<sup>4</sup>

(3)

<sup>=</sup> <sup>2</sup>*e*14*<sup>k</sup> <sup>q</sup>*<sup>4</sup> ε11 <sup>η</sup> (*<sup>k</sup>* <sup>2</sup> <sup>+</sup> *<sup>q</sup>*<sup>4</sup> 2) ..

( ) ( ) ( ) ( )

*Ae Ae Be Be C De C De E Fe E Fe A A BB CD CD EF EF*

2 2 4 4

*iq h iq h iq h iq h*

é ù ê ú - - +- +-

( ) ( ) ( ) ( )


*k*3 (*n*) ω *vt*

are equal to

ω *vl*

is the solution of biquadratic dispersive Eq. (8).

) is modulus of the wave vector of the longitudinal bulk acoustic wave, and *qn*≡*k*<sup>3</sup>

<sup>1</sup> <sup>4</sup> 4 16 .

) is the modulus of wave vector of the shear bulk acoustic wave, and *k*<sup>14</sup> <sup>=</sup> *<sup>e</sup>*<sup>14</sup>

is the electromechanical coupling coefficient (EMCC). Eigenvectors corresponding to a value

As a result, in this case, the determinant of the boundary conditions (Eq. (5)) is obtained as

2 24 4 ,




*iq h iq h iq h iq h*

ë û (15)

ë û (16)

For *SH*-waves, the characteristic dispersive equation similar to Eq. (8) can be written as

( ) ( ) η 2 2 2 2 2 2 22 11 44 1 3 0 1 3 14 1 3 ε ρ ω 4 0. *<sup>E</sup>* - +- +- = é ù *C k k k k ekk*

Solving Eq. (15) with respect to *k*3 at the parameter *k*≡*k*1, one can obtain the result:

<sup>2</sup> 1,2,3,4 2 2 22 2 22 24 3 14 14 14

<sup>2</sup> *t t k i k k kk k kk kk* é ù =± - - ± - + ê ú

( ) ( )

(*n*)

ε11 <sup>η</sup> *<sup>C</sup>*<sup>44</sup> *E*

(17)

(18)

$$\begin{aligned} \frac{\text{th}\left(\dot{iq}\_{z}h\right)}{\text{th}\left(\dot{iq}\_{z}h\right)} &= \frac{q\_{z}}{q\_{z}} \frac{\left[\boldsymbol{C}\_{44}^{\varepsilon}\varepsilon\_{11}^{3}\left(k^{2}+q\_{4}^{2}\right)+2\varepsilon\_{14}^{2}k^{2}\right]\left[\boldsymbol{v}\_{z1}^{\varepsilon\_{1}}\left(k^{2}-q\_{2}^{2}\right)+2\dot{v}\_{z}q\_{2}\text{th}\left(iq\_{2}h\right)\right]}{\left[\boldsymbol{C}\_{44}^{\varepsilon}\varepsilon\_{11}^{3}\left(k^{2}+q\_{2}^{2}\right)+2\varepsilon\_{14}^{2}k^{2}\right]\left[\boldsymbol{v}\_{z1}^{\varepsilon\_{1}}\left(k^{2}-q\_{4}^{2}\right)+2\dot{v}\_{z}q\_{4}\text{th}\left(iq\_{4}h\right)\right]},\\ \frac{\text{th}\left(iq\_{4}h\right)}{\text{th}\left(iq\_{2}h\right)} &= \frac{q\_{4}}{q\_{2}} \left[\frac{\boldsymbol{C}\_{44}^{\varepsilon}\varepsilon\_{11}^{3}\left(k^{2}+q\_{4}^{2}\right)+2\varepsilon\_{14}^{2}k^{2}}{\boldsymbol{C}\_{44}^{\varepsilon\_{1}}\varepsilon\_{11}^{3}\left(k^{2}+q\_{2}^{2}\right)+2\dot{v}\_{z}q\_{2}\text{th}\left(iq\_{2}h\right)}\right]}. \end{aligned} \tag{20}$$

It should be noted that in this case the *SH*-wave is also divided into symmetric and antisym‐ metric modes, which is possible only at the high symmetric directions of elastic wave propa‐ gation.

If we need to deposit a thin metal layer on the upper surface of the piezoelectric crystalline substrate without disturbing the mechanical boundary conditions, then the second row of the determinant (18) is changed, which leads to the following new dispersive equation for *SH*wave:

$$\begin{aligned} \frac{\text{th}\left(iq\_{q}h\right)}{\text{th}\left(iq\_{q}h\right)} &= \frac{q\_{q}}{q\_{1}} \left[ \frac{C\_{u}^{v}c\_{11}^{v}\left(k^{2}+q\_{1}^{2}\right)+2c\_{1}^{v}k^{2}\right] \left[c\_{11}^{v}\left(k^{2}-q\_{2}^{2}\right)+2ic\_{0}q\_{1}\text{th}\left(iq\_{q}h\right)\right]}{\text{e}\_{1}^{v}\left(C\_{u}^{v}c\_{11}^{v}\left(k^{2}+q\_{2}^{2}\right)+2c\_{11}^{v}k^{2}\right) \left[c\_{11}^{v}\left(k^{2}-q\_{q}^{2}\right)+2ic\_{0}q\_{1}\text{th}\left(iq\_{q}h\right)\right]} \\ &\times \left( \frac{1+\sum\_{1\mid h}^{v}\left(k^{2}-q\_{2}^{2}\right)\text{th}\left(iq\_{q}h\right)+2ic\_{0}q\_{2}}{\text{e}\_{11}^{v}\left(k^{2}-q\_{2}^{2}\right)\text{th}\left(iq\_{q}h\right)+2ic\_{0}q\_{2}} \right) \\ &+ \frac{\text{e}\_{1}^{v}\left(k^{2}-q\_{2}^{2}\right)\text{th}\left(iq\_{q}h\right)+2ic\_{0}q\_{2}}{\text{e}\_{11}^{v}\left(k^{2}-q\_{2}^{2}\right)\text{th}\left(iq\_{q}h\right)+2ic\_{0}q\_{2}} \end{aligned} \tag{21}$$

In this case, an additional factor on the right side is the dynamic electromechanical coupling coefficient, depending on the frequency/plate thickness, and taking into account the effect of metallization on the magnitude of the phase velocity of piezoactive *SH*-wave.

Another marginal case is the propagation of an elastic wave when interlayer interface has been taken into account. Let us consider the propagation of the Love wave (*SH*-wave in this case) along the [100] direction in (001) plane of the layered structure as "isotropic dielectric layer/ piezoelectric crystalline substrate of 23-point symmetry group." Characteristic equation for *SH*-waves is similar to Eq. (15). Equation (4) for the *SH*-wave in the form of partial waves can be written as

$$\begin{split} U\_{2} &= \left[ b^{(1)} \mathfrak{P}^{(1)} \exp \left( q\_{3}^{(1)} \mathbf{x}\_{3} \right) + b^{(2)} \mathfrak{P}^{(2)} \exp \left( q\_{3}^{(2)} \mathbf{x}\_{3} \right) \right] \exp \left[ i \left( k \mathbf{x}\_{l} - \alpha t \right) \right], \\ \Phi &= \left[ b^{(1)} \exp \left( q\_{3}^{(1)} \mathbf{x}\_{3} \right) + b^{(2)} \exp \left( q\_{3}^{(2)} \mathbf{x}\_{3} \right) \right] \exp \left[ i \left( k \mathbf{x}\_{l} - \alpha t \right) \right], \end{split} \tag{22}$$

where

$$\mathfrak{J}^{(1)} = \frac{U\_{\;2}}{\mathfrak{O}}\bigg|\_{\mathfrak{q}\_{\;j}^{(1)}} = \frac{\mathfrak{e}\_{11}^{(\mathrm{S})} \Big[k^{2} + \left(\mathfrak{q}\_{\mathrm{s}}^{(1)}\right)^{2}\Big]}{2\mathfrak{e}\_{14}^{(\mathrm{S})}k\mathfrak{q}\_{\;j}^{(1)}},\\\mathfrak{J}^{(2)} = \frac{U\_{\;2}}{\mathfrak{O}}\bigg|\_{\mathfrak{q}\_{\;j}^{(2)}} = \frac{\mathfrak{e}\_{11}^{(\mathrm{S})} \Big[k^{2} + \left(\mathfrak{q}\_{\mathrm{s}}^{(2)}\right)^{2}\Big]}{2\mathfrak{e}\_{14}^{(\mathrm{S})}k\mathfrak{q}\_{\;j}^{(2)}}\tag{23}$$

Here and after the superscript S is marked the substrate material constants. In an isotropic dielectric layer, due to the absence of the piezoelectric effect, the parameters of partial components are *k*<sup>3</sup> (1,2) = ± *is* and *k*<sup>3</sup> (3,4) <sup>=</sup> <sup>±</sup> *ik*, where *<sup>s</sup>* <sup>=</sup> <sup>ρ</sup><sup>0</sup> (L) ω2 *C*<sup>44</sup> (L) - *k* <sup>2</sup> . Here and after the superscript L is marked the layer material data. Consequently, the boundary conditions (6) can be written as

$$\begin{aligned} a^{(i)}\exp(ik\_3^{(1)}h) - a^{(i)}\exp(ik\_3^{(1)}h) &= 0, \\ b\left(a^{(s)}\left(-k\_1e\_{11}^{(1)} + \varepsilon\_o\right)\exp\left(ik\_3^{(s)}h\right) + a^{(s)}\left(k\_4e\_{11}^{(1)} + \varepsilon\_o\right)\exp\left(ik\_3^{(s)}h\right) = 0, \\ b\left(b^{(s)}\left(c\_{44}^{(s)}q\_1^{(1)}p\_1 + e\_{44}^{(s)}k\right) + b^{(i)}\left(C\_{44}^{(s)}q\_1^{(2)}p\_2 + e\_{44}^{(s)}k\right) + i\varepsilon C\_{44}^{(1)}\left(a^{(4)} - a^{(5)}\right)\right) &= 0, \\ b\left(b^{(i)}\left(c\_{14}^{(s)}kp\_1 - c\_{11}^{(s)}q\_1^{(1)}\right) + b^{(i)}\left(c\_{14}^{(s)}kp\_2 - c\_{11}^{(s)}q\_1^{(2)}\right) + i\varepsilon k\_{11}^{(1)}\left(a^{(3)} - a^{(6)}\right) &= 0, \\ b^{(i)}p\_1 + b^{(i)}p\_2 - a^{(5)} - a^{(4)} &= 0, \\ b^{(i)} + b^{(i)} - a^{(6)} - a^{(6)} &= 0, \\ p\_{1,2} &= \frac{c\_{11}^{(8)}\left[k^2 + \left(q\_1^{(1)}\right)^2\right]}{2\varepsilon\_4^{(8)}kq\_1^{(1)}}.\end{aligned} \tag{24}$$

Equating to zero the determinant of Eq. (24), one can get the dispersive equation for the Love wave propagating in the layered structure "isotropic dielectric layer/piezoelectric crystalline substrate" [13]:

$$\begin{aligned} &C\_{44}^{(1)}\text{ish}\left(sh\right) \\ &=\frac{AB}{\varepsilon\_{11}^{(8)}}\frac{\left[C\_{44}^{(8)}\varepsilon\_{11}^{(8)}\left(A^{2}+k^{2}\right)+2\left(\varepsilon\_{11}^{(8)}\right)^{2}k^{2}\right]-P\left[C\_{44}^{(8)}\varepsilon\_{11}^{(8)}\left(B^{2}+k^{2}\right)+2\left(\varepsilon\_{14}^{(8)}\right)^{2}k^{2}\right]}{B\left(A^{2}+k^{2}\right)-PA\left(B^{2}+k^{2}\right)},\end{aligned} \tag{25}$$

where *A*=*q*<sup>3</sup> (1) , *B* =*q*<sup>3</sup> (2) , *<sup>P</sup>* <sup>=</sup> *<sup>B</sup> <sup>A</sup>* ⋅ ε11 (S) (*k* <sup>2</sup> − *A* 2) + 2*iA*ε¯ ε11 (S) (*<sup>k</sup>* <sup>2</sup> <sup>−</sup> *<sup>B</sup>* 2) <sup>+</sup> <sup>2</sup>*iB*ε¯ , and ε¯ =ε<sup>11</sup> (L) *k* ε11 (L) *k* th(*kh* ) + ε<sup>0</sup> ε11 (L) *<sup>k</sup>* <sup>+</sup> <sup>ε</sup>0th(*kh* ) . Similar dispersive relation for Love waves (*SH*-waves) for a layered structure "isotropic layer/ piezoelectric substrate of hexagonal symmetry" has been obtained in Ref. [14].

#### **2.3. Analysis of anisotropy and dispersion of SAW parameters in "AlN/diamond" PLS**

where

170 Piezoelectric Materials

( )

components are *k*<sup>3</sup>

ï

ï

substrate" [13]:

where *A*=*q*<sup>3</sup>

( ) ( )

th

*C is sh*

( ) ( )

( )

11

(1) , *B* =*q*<sup>3</sup> (2) , *<sup>P</sup>* <sup>=</sup> *<sup>B</sup> <sup>A</sup>* ⋅ ε11 (S)

L 44

as

( )

= ± *is* and *k*<sup>3</sup>

(1,2)

( ) ( ) ( ) ( ) ( )

(3,4)

1 2 3 3

( ) ( ) ( ) ( ) ( ) ( )

exp exp 0,

3 14 2 3 3 5 L 3 6L 4 1 11 0 3 1 11 0 3 1 S1 S 2 S2 S L 4 3 44 3 1 14 44 3 2 14 44 1 S S1 2 S S2 L 5 6 14 1 11 3 14 2 11 3 11 1 2 34 1 2 125

*a ik h a ik h*

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

<sup>ì</sup> -+ + + = <sup>ï</sup>

*a k ik h a k ik h*

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

*k q*

<sup>2</sup> <sup>S</sup> <sup>2</sup> 1,2 11 3 1,2 S 1,2 14 3

í + + + + -=

ε ε exp ε ε exp 0,


ε ε ε 0,

6


*a*

é ù <sup>+</sup> ê ú ë û <sup>=</sup>

. <sup>2</sup>

Equating to zero the determinant of Eq. (24), one can get the dispersive equation for the Love wave propagating in the layered structure "isotropic dielectric layer/piezoelectric crystalline

( ) ( )

(L) *k* ε11 (L)

ε11 (L)

2 2 S S <sup>2</sup> 22 2 <sup>S</sup> S S 2 2 <sup>S</sup> 44 11 11 44 11 14

ε 2 ε ε 2 ε , <sup>ε</sup>

*C A k k PC B k k AB B A k PA B k* é ù é ù ++ - ++ ê ú ê ú ë û ë û <sup>=</sup> +- +

(*<sup>k</sup>* <sup>2</sup> <sup>−</sup> *<sup>B</sup>* 2) <sup>+</sup> <sup>2</sup>*iB*ε¯ , and ε¯ =ε<sup>11</sup>

0,

( ) ( ) ( )

*k* th(*kh* ) + ε<sup>0</sup>

*<sup>k</sup>* <sup>+</sup> <sup>ε</sup>0th(*kh* ) .

0,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

<sup>ï</sup> - + - + -= <sup>î</sup> + --= + - ( )

*b e kp q b e kp q ik a a*

*bp b p a a bb a*

ε

*<sup>p</sup> e kq*

( ) ( ) ( ) ( ) ( )

S 22 22

(*k* <sup>2</sup> − *A* 2) + 2*iA*ε¯

ε11 (S)

*b C q p e k b C q p e k isC a a*

<sup>β</sup> , β Φ Φ 2 2 *q q*

( )

Here and after the superscript S is marked the substrate material constants. In an isotropic dielectric layer, due to the absence of the piezoelectric effect, the parameters of partial

<sup>=</sup> <sup>±</sup> *ik*, where *<sup>s</sup>* <sup>=</sup> <sup>ρ</sup><sup>0</sup>

L is marked the layer material data. Consequently, the boundary conditions (6) can be written

<sup>11</sup> <sup>3</sup> <sup>11</sup> <sup>3</sup> 1 2 2 2

ε ε

*k q k q U U*

( )

(L) ω2 *C*<sup>44</sup> (L) - *k* <sup>2</sup>

2 2 S 1 2 2 S 2

S 1 S 2 14 3 14 3

éù é ù + + êú ê ú ëû ë û = = = = (23)

*e kq e kq*

( ) ( ) ( ) ( ) ( )

. Here and after the superscript

0,

(24)

(25)

Algorithm for calculating the elastic wave parameters in PLS was based on the method of partial waves, previously well adapted to define the SAW characteristics. To improve the accuracy of calculation, the normalization of Christoffel equations and boundary conditions was applied. Square of EMCC was determined by the relation

$$K^2 = 2\frac{\nu - \nu\_m}{\nu},\tag{26}$$

where *v* and *vm* are SAW phase velocities on the free or metallized surface of the piezoelectric film.

Using data on the elastic properties of diamond [15], and aluminum nitride (AlN) [16], the computer simulations of the SAW propagation in PLS "AlN/(111) diamond" were carried out with options of (100) or (001) on the AlN film orientation. Anisotropy of the Rayleigh wave phase velocity in the (111) plane of the diamond is relatively small, and the (001) plane of AlN film is isotropic with respect to the elastic properties; so, in PLS "(001) AlN/(111) diamond," the SAW propagation is actually happening as on the isotropic medium. In contrast, the "(100) AlN/(111) diamond" PLS gives us an example of noticeable anisotropy in the phase velocity and EMCC.

Determination of what SAW type will be excited in this case is possible, considering the tensor of piezoelectric coefficients for the 6*mm* point symmetry crystal AlN, which has a form in the special Cartesian coordinate system as

$$e'\_{1\boldsymbol{\lambda}} = \begin{pmatrix} e\_{3\boldsymbol{\lambda}} & e\_{3\boldsymbol{\lambda}} & e\_{3\boldsymbol{\lambda}} \ 0 & 0 & 0 \\ 0 & 0 & 0 \ 0 & 0 & e\_{1\boldsymbol{\lambda}} \\ 0 & 0 & 0 \ 0 & e\_{1\boldsymbol{\lambda}} & 0 \end{pmatrix} \tag{27}$$

Let us choose the SAW propagation X́2 along the [010] direction in the (100) plane of AlN film. AC electric field vector in accordance with the orientation of IDT will have the *E* =(0, *E*<sup>2</sup> ′, *E*<sup>3</sup> ′) components located in the sagittal plane. Components of mechanical stress can be found by the relation σ*ij* ' =*enij* ' *En* ' , where σ<sup>1</sup> ' =σ<sup>2</sup> ' =σ<sup>3</sup> ' =σ<sup>4</sup> ' =0, σ<sup>5</sup> ' =*e*15*E*<sup>3</sup> ' , and σ<sup>6</sup> ' =*e*15*E*<sup>2</sup> ' . Last component is responsible for pure *SH*-wave (Love wave) excitation. If we consider the SAW propagation along the X́ || [100] axis in the (100) plane of AlN film, it is fulfilled *E* =(*E* ′ 1, 0, *E* ′ 3 ), , and stress components are of the form: σ́1 = *e*33*É*1, σ́2 = *e*31*É*1, σ́3 = *e*31*É*1, σ́5 = *e*15*É*3, and σ́4 = σ́6 = 0. Components σ́1 and σ́<sup>5</sup> are responsible for the excitation of longitudinal and transverse components of the partial waves of the Rayleigh SAW; σ́2 and σ́3 are associated with parasitic BAW excitation. Consequently, for a given orientation, the possibility of piezoelectric *SH*-wave excitation is absent. In principle, this situation is analogous to the case of the (001) orientation of the AlN film, in which at any IDT location the Rayleigh SAW will only be excited.

**Figure 1.** Dispersion dependences of SAW phase velocities (a) and EMCCs (b) versus the *h* × *f* product for the "(001) AlN/(111) diamond" piezoelectric layered structure. Fast and slow bulk acoustic waves are designated as *FS* и *SS*, re‐ spectively. Curves *R*0 – *R*2 and *L*0, *L*1 are associated with the Rayleigh and Love surface acoustic waves.

Dispersion curves of SAW phase velocities and EMCCs for the "(001) AlN/(111) diamond" PLS depending on the *h* × *f* product are shown in **Figure 1**. Here, *h* is the AlN film thickness. Phase velocity of the fundamental *R*0 Rayleigh wave is changed from 10844 m/s (Rayleigh wave propagating in the (111) diamond plane at *h* = 0) up to 5775 m/s (Rayleigh wave propagating in the (001) plane of AlN film at *h* → ∞). Phase velocity of the lowest *L*<sup>0</sup> Love mode is changed from 11992 m/s (*SS* mode in diamond) up to 6511 m/s at *h* × *f* = 8000 m/s tending to *FS* phase velocity in AlN film, which is equal to 6398 m/s. As one can see, the only Rayleigh waves can be excited by piezoelectric effect because their EMCCs have non-zero values, but the excitement of *SH*-waves should be absent (**Figure 2b**). Maximal value of *K*<sup>2</sup> = 2.1 % with *h* × *f* = 2900 m/s will be observed for the *R*<sup>1</sup> mode which is often called as Sezawa wave. These dependences are qualitatively close to those obtained by the authors [17].

Dispersion curves of SAW phase velocities and EMCCs versus the *h* × *f* product for the "(100) AlN/(111) diamond" PLS are shown in **Figure 2**. In this case, there is a significant anisotropy of the elastic and piezoelectric properties of AlN film; therefore, it is necessary to specify a definite direction of wave propagation. Phase velocity of the *R*<sup>0</sup> mode is changed from 10787 up to 5884 m/s (Rayleigh wave propagating in the (100) plane of AlN film along the [010] direction at *h* → ∞). Phase velocity of the lowest *L*<sup>0</sup> mode is changed from 11992 up to 6282 m/ s, tending to *SS* phase velocity in AlN film, which is equal to 6172 m/s. In this direction only *SH*-waves can be excited by a piezoelectric effect. On the other hand, the excitation of Rayleigh waves should be absent (**Figure 2b**). Maximum value of *K*<sup>2</sup> = 0.76 % will be observed for the *L*0 mode at *h* × *f* = 2050 m/s. Obtained dependences are qualitatively close to the similar ones [18].

partial waves of the Rayleigh SAW; σ́2 and σ́3 are associated with parasitic BAW excitation. Consequently, for a given orientation, the possibility of piezoelectric *SH*-wave excitation is absent. In principle, this situation is analogous to the case of the (001) orientation of the AlN

**Figure 1.** Dispersion dependences of SAW phase velocities (a) and EMCCs (b) versus the *h* × *f* product for the "(001) AlN/(111) diamond" piezoelectric layered structure. Fast and slow bulk acoustic waves are designated as *FS* и *SS*, re‐

Dispersion curves of SAW phase velocities and EMCCs for the "(001) AlN/(111) diamond" PLS depending on the *h* × *f* product are shown in **Figure 1**. Here, *h* is the AlN film thickness. Phase velocity of the fundamental *R*0 Rayleigh wave is changed from 10844 m/s (Rayleigh wave propagating in the (111) diamond plane at *h* = 0) up to 5775 m/s (Rayleigh wave propagating in the (001) plane of AlN film at *h* → ∞). Phase velocity of the lowest *L*<sup>0</sup> Love mode is changed from 11992 m/s (*SS* mode in diamond) up to 6511 m/s at *h* × *f* = 8000 m/s tending to *FS* phase velocity in AlN film, which is equal to 6398 m/s. As one can see, the only Rayleigh waves can be excited by piezoelectric effect because their EMCCs have non-zero values, but the

*h* × *f* = 2900 m/s will be observed for the *R*<sup>1</sup> mode which is often called as Sezawa wave. These

Dispersion curves of SAW phase velocities and EMCCs versus the *h* × *f* product for the "(100) AlN/(111) diamond" PLS are shown in **Figure 2**. In this case, there is a significant anisotropy of the elastic and piezoelectric properties of AlN film; therefore, it is necessary to specify a definite direction of wave propagation. Phase velocity of the *R*<sup>0</sup> mode is changed from 10787 up to 5884 m/s (Rayleigh wave propagating in the (100) plane of AlN film along the [010] direction at *h* → ∞). Phase velocity of the lowest *L*<sup>0</sup> mode is changed from 11992 up to 6282 m/ s, tending to *SS* phase velocity in AlN film, which is equal to 6172 m/s. In this direction only *SH*-waves can be excited by a piezoelectric effect. On the other hand, the excitation of Rayleigh

= 2.1 % with

spectively. Curves *R*0 – *R*2 and *L*0, *L*1 are associated with the Rayleigh and Love surface acoustic waves.

excitement of *SH*-waves should be absent (**Figure 2b**). Maximal value of *K*<sup>2</sup>

dependences are qualitatively close to those obtained by the authors [17].

film, in which at any IDT location the Rayleigh SAW will only be excited.

172 Piezoelectric Materials

**Figure 2.** Dispersion dependences of SAW phase velocities (a) and EMCCs (b) versus the *h* × *f* product for the "(100) AlN/(111) diamond" PLS. SAW propagation was chosen along the [010] direction of AlN film and coincided with the direction 112 ¯ in (111) diamond plane.

**Figure 3.** Anisotropy of SAW phase velocity (a) and EMCC (b) in the "(001) AlN/(111) diamond" PLS at different val‐ ues of *h* × *f* (m/s): (1) 1000; (2) 3000; (3) 5000.

**Figure 3** represents the anisotropy of SAW parameters in the "(001) AlN/(111) diamond" PLS. A set of curves is associated with data for the three values of the *h* × *f* product. Investigated structure shows the highest EMCC for the Rayleigh-type waves and low anisotropy of the phase velocity due to weak anisotropy of elasticity in the (111) diamond plane. Thus, highvelocity Sezawa mode *R*1 has the maximal value *K*<sup>2</sup> = 2.1 %, and the phase velocity of 11531 m/ s at *h* × *f* = 3000 m/s.

**Figure 4.** Anisotropy of SAW phase velocity (a), EMCC (b), and PFA (c) in the "(100) AlN/(111) diamond" PLS at dif‐ ferent values of *h* × *f* (m/s): (1) 1000; (2) 3000; (3) 5000. Angle *ψ* was measured from the [010] up to [001] direction in the (100) plane of the AlN film. The [010] direction of AlN film coincided with the 112 ¯ direction in (111) diamond plane.

**Figure 4** shows the anisotropy of SAW parameters in the "(100) AlN/(111) diamond" PLS. As follows from **Figure 4b**, best EMCC values of 1.6 and 0.7% have a fundamental Rayleigh mode *R*0, and Sezawa mode *R*1 at *h* × *f* = 5000 m/s in the [001] propagation direction of AlN film (*ψ* = 90 °). Sezawa mode has also the greatest value of the phase velocity of 12290 m/s (**Figure 4a**). Note that in this direction a propagation of the pure modes will be realized both for the Rayleigh and *SH*-type waves (**Figure 4c**), because power flow angle (PFA) tends to zero, but only Rayleigh modes can be excited due to the AlN piezoelectric effect. Love wave has a maximal EMCC *K*<sup>2</sup> = 1.35 % in the direction *ψ* = 56 °, but it is no longer a pure mode.
