**4. Modeling of microwave acoustic properties of "Me1/AlN/Me2/diamond" piezoelectric layered structure**

#### **4.1. Equivalent circuit and frequency dependences of equivalent parameters for «Me1/AlN/ Me2/diamond» PLS**

HBAR's equivalent parameters are close concerned with specified equivalent scheme of device and are important in view of HBAR modeling and application, for example, the AFC calcula‐ tion, matching of devices, etc. PLS equivalent scheme introduced in Ref. [6] is represented in **Figure 21**.

$$\begin{aligned} C\_o &\approx -\frac{1}{\text{co}\_s \text{Im} Z\_{11}}, \\ R\_s &= \max \text{Re} Z\_e \left( f\_{p,n} \right), \\ k\_n^2 &= \frac{R\_n C\_0 \text{co}\_{p,n}}{Q\_n}, \\ C\_n &= \frac{C\_0}{k\_n^2}, \\ L\_n &= \frac{k\_n^2}{\text{co}\_{p,n}^2 C\_0} = \frac{1}{\text{co}\_{p,n}^2 C\_s}. \end{aligned} \tag{36}$$

Here *C*0 and *Cn* are static and dynamic capacities, *Rn* is loss of resistance, *kn* is the electrome‐ chanical coupling coefficient, *Ln* is dynamic inductance, ω*<sup>p</sup>*,*<sup>n</sup>* = 2π*fp*,*<sup>n</sup>* is angular frequency of parallel resonance.

Frequency dependence of equivalent parameters for the 1a, 1b, and 1c resonators (PLS #3) is represented in **Figure 22**. As one can see, there is a complex frequency dependence of equiv‐ alent parameters which requires an accurate choice of operational frequency bands.

**Figure 21.** PLS equivalent scheme.

**Figure 22.** Frequency dependence of equivalent parameters for the 1a, 1b, and 1c resonators (PLS #3, see **Figure 7** and Section 3.1).

#### **4.2. Peculiarities of PLS acoustic wave excitation by thin film piezoelectric transducer**

Both *Q* and *Q* × *f* measured for a lot of PLSs have nonmonotonic frequency dependence with a number of local maximums and minimums. Such complex dependence can be explained in terms of form-factor *m*, which was introduced in Ref. [29] for the TFPT conjugated with the acoustic line. It was shown in Ref. [29] that the frequency dependence of the acoustic power *W* radiated by the TFPT into the substrate can be considered by the relation: *W* ~ |*m*|2 . In Ref. [21], it was shown that such approach is also valid for the "Me1/piezoelectric/Me2/substrate" PLS. A simple analytic expression of the form-factor *m* for PLS without top electrode [21,29] can be written as

**Figure 21.** PLS equivalent scheme.

188 Piezoelectric Materials

Section 3.1).

**Figure 22.** Frequency dependence of equivalent parameters for the 1a, 1b, and 1c resonators (PLS #3, see **Figure 7** and

$$m = \frac{\cos\phi\_p \cos\phi\_M - \frac{Z\_p}{Z\_M}\sin\phi\_p \sin\phi\_M + i\left(\frac{Z\_M}{Z\_p}\cos\phi\_p \sin\phi\_M + \frac{Z\_p}{Z}\sin\phi\_p \cos\phi\_M\right)}{\sin^2\frac{\phi\_p}{2}}.\tag{37}$$

Here *ZP*, *ZM*, and *Z* are the acoustic impedances of the piezoelectric film, inner metal electrode, and substrate, respectively; φ*M* = *kMdM* and φ*P* = *kPdP* are the phase shifts inserted by the metal film and piezoelectric layer; *kM* and *kP* are the wave vectors in those layers; *dM* and *dP* are thicknesses of these layers. However, Eq. (37) should be considerably modified in order to take into account the top electrode. Unfortunately, such equation could not be written in a compact form, but the proper matrix system of equations was represented in Ref. [21].

**Figure 23.** Frequency dependence of form-factor |*m*|2 , Re(*m*), and Im(*m*): (a) calculated data, and (b) comparison with experimentally obtained *Q* × *f* product for the PLS #3 (see **Figure 7** and Section 3.1).

It is easy to show that as a rule the minimums of |*m*|2 should be associated with maximumradiated acoustic power, while maximums of form-factor should result in low *W* or even the absence of an acoustic signal. A number of form-factor calculations were done by our special computer software. As one can see from **Figure 23**, there is a strong correlation between the frequency dependences of *Q* (*Q* × *f*) and |*m*|2 = *f*(ω), as well as with the Re(*m*) and Im(*m*) functions. It can be easily seen that frequency bands with high *Q* (*Q* × *f*) should be associated with minimal values of the Im(*m*), low *Q* (*Q* × *f*) values correlate with minimal Re(*m*), and low TFPT effectiveness coincides with maximal |*m*|2 . Note that Re(*m*) minimums arise at frequency bands concerned with the ~ *p*(λ*AlN*/4) resonances of AlN film (*p* is the odd integer). So the acoustoelectronic device will be more efficient at such operating frequencies where the flat regions with minimal values of the |*m*|2 function are realized. As a result, such preliminary computer analysis makes possible defining the proper operating frequencies before construct‐ ing a device.

#### **4.3. 2D FEM simulation results of acoustic wave propagation in diamond-based PLS**

The 2D FEM simulations were carried out with the use of Comsol Multiphysics Simulation Software. Material data on the density, elastic constants, acoustic attenuation, etc. for all the layers and diamond substrate were taken from Refs. [16, 21, 30]. Such PLS parameters and acoustic processes as distribution of elastic displacement fields within all the parts of PLS (cross-sections of substrate and TFPT), AFR modeling, calculation of the wavelengths and phase velocities of acoustic modes of different types, identification of its types, etc. have been studied in detail. Symmetrical boundary conditions on the lateral borders of diamond substrate were used in order to satisfy the condition of zero normal displacement components as (*n* <sup>→</sup> , *U* <sup>→</sup>)=0, where *<sup>n</sup>* <sup>→</sup> is a unit vector of the normal to the lateral border, and *U* <sup>→</sup> is a unit vector of the wave elastic displacement. Visualization of elastic displacement fields gave us an instrument of analysis of acoustic wave excitation because all the waves propagating in lateral directions should be reflected under the condition *w* = *m*(λ/2), where *w* is a substrate width, *λ* is a wavelength, and *m* is an integer. In this 2D case, as a result of wave reflection, a lot of enabled acoustic waves such as symmetrical (*Sn*) and antisymmetrical (*An*) Lamb waves of different orders, and Rayleigh waves could be observed in a given frequency band. Overtones of longitudinal BAW, which were of great importance in a practical sense have been investi‐ gated in a wide frequency band in order to establish the energy-trapping effect in diamondbased PLS. Calculated layered structure was chosen as to be close to the experimentally studied one as PLS #3. Widths of top Al electrode, AlN layer, and Mo bottom electrode were equal to 400 μm, while diamond's widths were taken as 1000 or 1100 μm. Thicknesses of Al, AlN, Mo layers, and diamond substrate were equal to 164 nm, 624 nm, 169 nm, and 392 μm, respectively. During the calculation by the "Eigenfrequency" option, a lot of sequentially excited acoustic resonant modes have been observed, and the main attention was paid on the patterns of mechanical displacements, which were associated with one or the other type of acoustic wave. Because the most waves in PLS have a dispersion of phase velocities, it was important to study its frequency dependences. The form of more convenient presentation of calculated results was the YX cross-section of a sample (**Figure 24**). Wave propagation can be observed along the X-axis (Rayleigh and Lamb waves) or along the Y-axis (mainly longitudinal acoustic waves).

**Figure 24.** Typical scheme of diamond-based PLS.

It is easy to show that as a rule the minimums of |*m*|2 should be associated with maximumradiated acoustic power, while maximums of form-factor should result in low *W* or even the absence of an acoustic signal. A number of form-factor calculations were done by our special computer software. As one can see from **Figure 23**, there is a strong correlation between the

functions. It can be easily seen that frequency bands with high *Q* (*Q* × *f*) should be associated with minimal values of the Im(*m*), low *Q* (*Q* × *f*) values correlate with minimal Re(*m*), and low

bands concerned with the ~ *p*(λ*AlN*/4) resonances of AlN film (*p* is the odd integer). So the acoustoelectronic device will be more efficient at such operating frequencies where the flat

computer analysis makes possible defining the proper operating frequencies before construct‐

The 2D FEM simulations were carried out with the use of Comsol Multiphysics Simulation Software. Material data on the density, elastic constants, acoustic attenuation, etc. for all the layers and diamond substrate were taken from Refs. [16, 21, 30]. Such PLS parameters and acoustic processes as distribution of elastic displacement fields within all the parts of PLS (cross-sections of substrate and TFPT), AFR modeling, calculation of the wavelengths and phase velocities of acoustic modes of different types, identification of its types, etc. have been studied in detail. Symmetrical boundary conditions on the lateral borders of diamond substrate were used in order to satisfy the condition of zero normal displacement components

<sup>→</sup> is a unit vector of the normal to the lateral border, and *U*

of the wave elastic displacement. Visualization of elastic displacement fields gave us an instrument of analysis of acoustic wave excitation because all the waves propagating in lateral directions should be reflected under the condition *w* = *m*(λ/2), where *w* is a substrate width, *λ* is a wavelength, and *m* is an integer. In this 2D case, as a result of wave reflection, a lot of enabled acoustic waves such as symmetrical (*Sn*) and antisymmetrical (*An*) Lamb waves of different orders, and Rayleigh waves could be observed in a given frequency band. Overtones of longitudinal BAW, which were of great importance in a practical sense have been investi‐ gated in a wide frequency band in order to establish the energy-trapping effect in diamondbased PLS. Calculated layered structure was chosen as to be close to the experimentally studied one as PLS #3. Widths of top Al electrode, AlN layer, and Mo bottom electrode were equal to 400 μm, while diamond's widths were taken as 1000 or 1100 μm. Thicknesses of Al, AlN, Mo layers, and diamond substrate were equal to 164 nm, 624 nm, 169 nm, and 392 μm, respectively. During the calculation by the "Eigenfrequency" option, a lot of sequentially excited acoustic resonant modes have been observed, and the main attention was paid on the patterns of mechanical displacements, which were associated with one or the other type of acoustic wave. Because the most waves in PLS have a dispersion of phase velocities, it was important to study its frequency dependences. The form of more convenient presentation of calculated results was the YX cross-section of a sample (**Figure 24**). Wave propagation can be observed along

**4.3. 2D FEM simulation results of acoustic wave propagation in diamond-based PLS**

= *f*(ω), as well as with the Re(*m*) and Im(*m*)

. Note that Re(*m*) minimums arise at frequency

<sup>→</sup> is a unit vector

function are realized. As a result, such preliminary

frequency dependences of *Q* (*Q* × *f*) and |*m*|2

TFPT effectiveness coincides with maximal |*m*|2

regions with minimal values of the |*m*|2

ing a device.

190 Piezoelectric Materials

as (*n* <sup>→</sup> , *U*

<sup>→</sup>)=0, where *<sup>n</sup>*

**Figure 25.** Displacement fields of some modes obtained by 2D FEM simulation for diamond-based PLS #3: (a) third overtone of the *L* mode at ~67 MHz; (b) the *A*0 mode at resonant frequency ~8.03 MHz; (c) the *R* mode at resonant frequency ~54.6 MHz; (d) the *S*0 mode at resonant frequency ~17.28 MHz; (e) the *S*2*<sup>l</sup>* Lamb mode near the critical fre‐ quency (*fcr* = 68.86 MHz); (f) the *S*2*<sup>t</sup>* Lamb mode near the critical frequency (*fcr* = 63.84MHz); (g) the *A*4*<sup>l</sup>* Lamb mode; and (h) *A*4*<sup>t</sup>* Lamb mode. Red color is associated with Y-component of elastic displacement in vertical direction up, and blue color defines such value in vertical direction down. Elastic displacements are presented in a magnified scale.

In **Figure 25**, the patterns of elastic displacements for some acoustic waves are presented. One can clearly see the arrangement of displacement vectors designated by arrows. Color graphics serves, knowing the displacement along the Y-direction (up or down) and its magnitude. For example, if we take into account Rayleigh wave displacements (**Figure 25c**), there are ten λ*R*/2, which are placed along the width of the diamond substrate (1000 μm); so, λ*R* = 200 μm at 54.613 MHz. As a result, the phase velocity of Rayleigh wave propagating on the diamond surface is equal to 10922.6 m/s. In a similar manner, the phase velocities for all observed eigenmodes have been calculated.

Effect of energy trapping was observed in the conventional piezoelectric resonators and was explained by authors [31] as a total internal reflection of acoustic beam on the vertical borders within the resonator's aperture. Such effect was observed in HBARs too [32], but it has a more complex nature. As an example, the appearance of energy trapping in the PLS #3 is represented in **Figure 26**. It should be noted that realization of energy trapping is strongly concerned with the BAW wavelength, lateral dimensions, substrate thickness, and TFPT aperture. In the case of PLS #3, the energy trapping regime in the steady state was established, beginning the 12th BAW overtone at frequencies above ~268 MHz.

**Figure 26.** Energy trapping in the PLS #3: (a) energy trapping realized for 34th BAW overtone (~759 MHz); (b) absence of energy trapping on 3rd overtone (~67 MHz).

#### **4.4. Identification, selection and classification of acoustic waves of different types**

In **Figure 25**, the patterns of elastic displacements for some acoustic waves are presented. One can clearly see the arrangement of displacement vectors designated by arrows. Color graphics serves, knowing the displacement along the Y-direction (up or down) and its magnitude. For example, if we take into account Rayleigh wave displacements (**Figure 25c**), there are ten λ*R*/2, which are placed along the width of the diamond substrate (1000 μm); so, λ*R* = 200 μm at 54.613 MHz. As a result, the phase velocity of Rayleigh wave propagating on the diamond surface is equal to 10922.6 m/s. In a similar manner, the phase velocities for all observed eigenmodes

Effect of energy trapping was observed in the conventional piezoelectric resonators and was explained by authors [31] as a total internal reflection of acoustic beam on the vertical borders within the resonator's aperture. Such effect was observed in HBARs too [32], but it has a more complex nature. As an example, the appearance of energy trapping in the PLS #3 is represented in **Figure 26**. It should be noted that realization of energy trapping is strongly concerned with the BAW wavelength, lateral dimensions, substrate thickness, and TFPT aperture. In the case of PLS #3, the energy trapping regime in the steady state was established, beginning the 12th

**Figure 26.** Energy trapping in the PLS #3: (a) energy trapping realized for 34th BAW overtone (~759 MHz); (b) absence

have been calculated.

192 Piezoelectric Materials

BAW overtone at frequencies above ~268 MHz.

of energy trapping on 3rd overtone (~67 MHz).

PLS has a sophisticated acoustic spectrum. Generally speaking there are three normal bulk acoustic waves propagating along any crystalline direction within a substrate: Rayleigh waves on free top and bottom substrate surfaces; symmetrical and antisymmetrical Lamb plate waves when the thickness of substrate is comparable to the wavelength. Presence of piezoelectric layer gives a possibility of excitation not only to bulk waves in vertical direction, but also to modified dispersive Rayleigh waves, including Sezawa wave with highest phase velocity, and Love waves, belonging to the *SH*-wave class. Excitation Rayleigh or *SH* waves separately will be defined as an occurrence of appropriate piezoelectric constants at the given configuration of electrodes and orientation of piezoelectric film (see Section 2.3). Mass loading of TFPT results in "softening" of the mutual region "substrate + TFPT" and gives a useful possibility of energy trapping for longitudinal BAW in the vertical direction, so that the BAW energy is localized within the TFPT aperture under certain conditions. Note that within TFPT region, the SAW phase velocity will have a smaller value than the same at the free diamond surface. Fine interferometric effects could be observed as a consequence of acoustic wave reflection from lateral borders of the substrate or TFTP.

Type and order of the acoustic modes were defined, taking into account the directions of elastic displacement vectors, location, and the number of homogeneous areas of elastic displacements along vertical and horizontal axes when the fields of elastic displacements analogous to **Figure 25** have been analyzed.

Dispersive Lamb waves are of special interest because there are a lot of types and orders of dispersive branches for its phase velocities. In order to distinguish Lamb modes, let us remember that the critical wavelengths (or critical frequencies) for each Lamb mode should be defined [33] as

$$\begin{aligned} h &= \frac{\lambda\_\iota}{2}, \frac{3\lambda\_\iota}{2}, \frac{5\lambda\_\iota}{2}, \dots \\ h &= \lambda\_\iota, 2\lambda\_\iota, 3\lambda\_\iota, \dots \\ h &= \lambda\_\iota, 2\lambda\_\iota, 3\lambda\_\iota, \dots \\ h &= \frac{\lambda\_\iota}{2}, \frac{3\lambda\_\iota}{2}, \frac{5\lambda\_\iota}{2}, \dots \end{aligned} \tag{38}$$

where *h* is the plate thickness, *λ<sup>l</sup>* and *λ<sup>t</sup>* are the wavelengths of longitudinal and transverse (shear) bulk acoustic waves, respectively. Note that at critical frequency, each Lamb mode will degenerate into a standing wave whose phase velocity will tend to infinity and elastic displacement field will correspond to either a longitudinal or transverse type. So, at the frequencies below a critical one, a given Lamb mode should not exist. Using Eq. (38), one can calculate the critical frequencies for each Lamb wave excited in a plate and define the point of creation of such mode. Note that PLS critical frequencies could weakly differ from the ones for pure plate. Now, the necessity for distinguishing between Lamb modes of the same type and order has been recognized. It would be a good thing to highlight this fact as a mode creation's point, and as a result, here and after, the lower index "*l*" or "*t*" will be used to designate not only critical frequencies, but also the dispersive curves calculated for such specified Lamb modes. Usually, such differences had not been taken into account, and, in our opinion, some mistakes of classification of the Lamb mode spectrum were previously made.

### **4.5. Phase velocity dispersion curves and critical frequencies for acoustic waveguide Lamb modes**

Dispersive properties of Lamb waves are important in a physical and practical sense. Based on 2D FEM simulation of PLS #3, all the possible modes have been observed within the 0– 250 MHz band, than the mode's identification and classification have been executed [34]. As a result, the dispersion curves for the phase velocities for a lot of Lamb modes have been obtained (**Figure 27**). As one can see, the phase velocities of the *A*0 and *S*0 modes converge to the phase velocity of Rayleigh wave at high frequency. It is suitable to say that such *A*<sup>0</sup> and *S*<sup>0</sup> modes have a broad application in acoustic nondestructive control devices, sensors, etc. Note that the conditions of Rayleigh wave excitation are fulfilled at both the substrate surfa‐ ces, but the effective phase velocity on the top surface should be lower due to the TFPT loading.

Phase velocities of all other *An* and *Sn* (*n* ≥ 1) modes have a high dispersion, and at high frequency finally converge to the phase velocity of shear BAW propagating along the [010] direction of the diamond. Super-high magnitudes of phase velocities of Lamb waves near the creation points have no physical meaning and can be treated as standing longitudinal/ transverse modes.

Analyzing a set of dispersive curves, one can notice that the difference between Lamb modes with respect to a mode creation's point (e.g., *Anl* vs *Ant*, or *Snl* vs *Snt*, etc.) has an important meaning, and such features are necessary for detailed explanation of Lamb wave spectrum in a wide sense.

Taking into account Eq. (38), one can see that critical frequencies for *Snl* and *Anl* plate modes (*n* ≥ 1) coincide with the resonant frequencies of odd and even overtones of longitudinal bulk acoustic wave, respectively. For example, such relations should be fulfilled as *fcr*(*S*1*<sup>l</sup>* ) = *fr*(*L*1), *fcr*(*S*2*<sup>l</sup>* ) = *fr*(*L*3), …, *fcr*(*A*1*<sup>l</sup>* ) = *fr*(*L*2), *fcr*(*A*2*<sup>l</sup>* ) = *fr*(*L*4), etc. As a result, this fact can also be used for Lamb mode identifying. But it should be noted that real critical frequencies for Lamb waves propa‐ gated in PLS studied differ slightly from that predicted by Eq. (38) due to the influence of mass loading as well as phase shifts arising as a consequence of deposition of thin-film layers on the top of substrate.

In the similar manner the dispersive curves of phase velocities of Lamb waves can be obtained at other frequency bands and will be useful for preliminary analysis of design of advanced acoustoelectronics devices.

**Figure 27.** Frequency dependences of phase velocity for diamond-based piezoelectric layered structure PLS #3.

## **5. Conclusion**

creation's point, and as a result, here and after, the lower index "*l*" or "*t*" will be used to designate not only critical frequencies, but also the dispersive curves calculated for such specified Lamb modes. Usually, such differences had not been taken into account, and, in our opinion, some mistakes of classification of the Lamb mode spectrum were previously made.

**4.5. Phase velocity dispersion curves and critical frequencies for acoustic waveguide**

Dispersive properties of Lamb waves are important in a physical and practical sense. Based on 2D FEM simulation of PLS #3, all the possible modes have been observed within the 0– 250 MHz band, than the mode's identification and classification have been executed [34]. As a result, the dispersion curves for the phase velocities for a lot of Lamb modes have been obtained (**Figure 27**). As one can see, the phase velocities of the *A*0 and *S*0 modes converge to the phase velocity of Rayleigh wave at high frequency. It is suitable to say that such *A*<sup>0</sup> and *S*<sup>0</sup> modes have a broad application in acoustic nondestructive control devices, sensors, etc. Note that the conditions of Rayleigh wave excitation are fulfilled at both the substrate surfa‐ ces, but the effective phase velocity on the top surface should be lower due to the TFPT

Phase velocities of all other *An* and *Sn* (*n* ≥ 1) modes have a high dispersion, and at high frequency finally converge to the phase velocity of shear BAW propagating along the [010] direction of the diamond. Super-high magnitudes of phase velocities of Lamb waves near the creation points have no physical meaning and can be treated as standing longitudinal/

Analyzing a set of dispersive curves, one can notice that the difference between Lamb modes with respect to a mode creation's point (e.g., *Anl* vs *Ant*, or *Snl* vs *Snt*, etc.) has an important meaning, and such features are necessary for detailed explanation of Lamb wave spectrum in

Taking into account Eq. (38), one can see that critical frequencies for *Snl* and *Anl* plate modes (*n* ≥ 1) coincide with the resonant frequencies of odd and even overtones of longitudinal bulk

mode identifying. But it should be noted that real critical frequencies for Lamb waves propa‐ gated in PLS studied differ slightly from that predicted by Eq. (38) due to the influence of mass loading as well as phase shifts arising as a consequence of deposition of thin-film layers on the

In the similar manner the dispersive curves of phase velocities of Lamb waves can be obtained at other frequency bands and will be useful for preliminary analysis of design of advanced

) = *fr*(*L*4), etc. As a result, this fact can also be used for Lamb

) = *fr*(*L*1),

acoustic wave, respectively. For example, such relations should be fulfilled as *fcr*(*S*1*<sup>l</sup>*

) = *fr*(*L*2), *fcr*(*A*2*<sup>l</sup>*

**Lamb modes**

194 Piezoelectric Materials

loading.

transverse modes.

a wide sense.

top of substrate.

) = *fr*(*L*3), …, *fcr*(*A*1*<sup>l</sup>*

acoustoelectronics devices.

*fcr*(*S*2*<sup>l</sup>*

A lot of "Al/AlN/Mo/(100) diamond" PLSs have been studied both theoretically and experi‐ mentally within a wide frequency band of 0.5–10 GHz. At first time, the highest among known material quality parameter *Q* × *f* ~ 1014 Hz for the IIa-type synthetic diamond at operational frequency ~10 GHz has been found. Analyzing the elastic displacement fields of PLS obtained by 2D FEM simulation, the Lamb modes as well as other types of elastic waves have been identified. Dispersive curves for phase velocities of all the acoustic waves observed have been plotted for the studied PLSs in the frequency range 0–250 MHz, and it is easy to expand such approach up to a higher frequency band. It has been established that 2D FEM visualization of the elastic displacement fields gives a fine possibility to an accurate study of the fundamental properties of elastic waves of different types. Analyzing the obtained data, one should note that the modified classification of Lamb modes in regard with its creation point as a longitu‐ dinal or transverse standing wave at critical frequency should be additionally introduced. Results on UHF acoustic attenuation of IIa-type synthetic single crystalline diamond have been presented and discussed in terms of Akhiezer and Landau–Rumer mechanisms of phonon– phonon interaction. It was found that the frequency of transformation of Akhiezer's mecha‐ nism into Landau–Rumer's was estimated as ~1 GHz in the IIa-type synthetic diamond at room temperature. As a result, single crystalline diamond will be a promising substrate for acous‐ toelectronic devices when operating frequency should be higher than the units of GHz, because the UHF acoustic attenuation in diamond in comparison with other commonly used crystals will be considerably lower.
