**3. Experimental Investigations of Acoustic Wave Excitation and Propagation in Diamond Based PLSs**

#### **3.1. Objects of Investigation**

The IIa-type synthetic diamond single crystals grown by HPHT method at the Technological Institute for Superhard and Novel Carbon Materials were used as substrates for the studied PLSs. All the substrate specimens were double-side polished up to the roughness *Ra* lower than 15 nm on 10 × 10 μm2 surface controlled by AFM method. Metal electrodes and AlN piezoelectric films were deposited by magnetron sputtering equipment AJA ORION 8. A preferred choice of Mo as a bottom electrode is explained by a good accordance between acoustical impedances of diamond and Mo. It can be seen by the X-ray diffraction pattern for the test sample AlN/Mo/glass (**Figure 5**) that the AlN film has the preferred orientation (002), and the full-width at half-maximum for this reflection is 0.213°. The X-ray diffraction meas‐ urements were performed by Empyrean (Panalytic) equipment. **Figure 5** shows SEM images of the surface and cross-section of the AlN/Mo/Si test sample. One can see the surface mor‐ phology of the AlN film with a crystallite size about 30–100 nm. The SEM investigations were performed by the JSM-7600F (JEOL) high-resolution scanning electron microscope. The PLS fabrication technology was in detail described in Ref. [19]. Electrode structures with a specified topology were deposited using both masks and photolithography by the Heidelberg μPG 101 equipment. Explosive photolithography was necessary to form a specified AlN film topology. The thickness of deposed films varied within 150–200 nm for top electrode, 0.4–5.5 μm for AlN piezoelectric layer, and 150–200 nm for bottom electrode. The set of the studied PLSs is represented in **Table 1**.

**Figure 5.** (a) X-ray diffraction pattern of AlN/Mo/glass test sample; (b) SEM images of AlN surface (magnification ×100,000); (c) SEM cross-section image (magnification ×50,000) of AlN/Mo/Si test sample.


**Table 1.** Set of studied PLSs.

**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

**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

The IIa-type synthetic diamond single crystals grown by HPHT method at the Technological Institute for Superhard and Novel Carbon Materials were used as substrates for the studied PLSs. All the substrate specimens were double-side polished up to the roughness *Ra* lower

piezoelectric films were deposited by magnetron sputtering equipment AJA ORION 8. A preferred choice of Mo as a bottom electrode is explained by a good accordance between acoustical impedances of diamond and Mo. It can be seen by the X-ray diffraction pattern for the test sample AlN/Mo/glass (**Figure 5**) that the AlN film has the preferred orientation (002), and the full-width at half-maximum for this reflection is 0.213°. The X-ray diffraction meas‐ urements were performed by Empyrean (Panalytic) equipment. **Figure 5** shows SEM images of the surface and cross-section of the AlN/Mo/Si test sample. One can see the surface mor‐ phology of the AlN film with a crystallite size about 30–100 nm. The SEM investigations were performed by the JSM-7600F (JEOL) high-resolution scanning electron microscope. The PLS

surface controlled by AFM method. Metal electrodes and AlN

**3. Experimental Investigations of Acoustic Wave Excitation and**

= 1.35 % in the direction *ψ* = 56 °, but it is no longer a pure mode.

¯ direction in (111) diamond plane.

(100) plane of the AlN film. The [010] direction of AlN film coincided with the 112

maximal EMCC *K*<sup>2</sup>

174 Piezoelectric Materials

**3.1. Objects of Investigation**

than 15 nm on 10 × 10 μm2

**Propagation in Diamond Based PLSs**

The influence of area size and the shape of the electrodes on the signal quality was studied. Serial number of resonator's notation is associated with the shape, while the letter represents the area of the top electrode. Here #"1" means a pentagon form, #"2"–irregular rectangle, and #"3"–circular form. The area of electrode "a" is 40000, "b"–22500, and "c"–10000 μm2 . All these 1a–3c electrodes are associated with different HBARs with the same AlN and metal layers.

### **3.2. UHF study of acoustic wave excitation and propagation in piezoelectric layered structures**

Microwave studies of PLSs were carried out by equipment (**Figure 6**) comprising the E5071C network analyzer (300 kHz–20 GHz) and the M-150 probe station. The experiments were fulfilled in the reflection mode with a test device connected by ACP40-A-SG-500 probe (the distance between tips was equal to 500 μm). **Figure 7** shows the photograph of one of the studied samples (PLS #3).

**Figure 6.** Experimental setup: E5071C network analyzer, M-150 probe station, and studied diamond-based PLS.

**Figure 7.** General view of the PLS#3 included nine acoustic resonators.

#### **3.3. PLS microwave acoustic properties measuring: phase velocities, quality factor and quality parameter, frequency and temperature dependences**

PLS gives us a unique possibility to investigate the acoustic properties within a wide frequency band from several MHz up to tens of GHz: from fundamental λ/2 resonance (*n* = 1) up to UHF overtones (*n* > 1000). Really, in the successful observation, all the overtones will depend on the quality and thickness of TFPT, and even more the acoustic attenuation in the substrate as well as the quality of their preparation. **Figure 8** represents frequency dependence of *S*<sup>11</sup> parameter (reflection coefficient) measured close to first acoustic overtone (*f*1*<sup>r</sup>* = 8.39543 MHz) of the PLS #1. The quality factor *Q* of an oscillator can be measured as

**3.2. UHF study of acoustic wave excitation and propagation in piezoelectric layered**

**Figure 6.** Experimental setup: E5071C network analyzer, M-150 probe station, and studied diamond-based PLS.

**3.3. PLS microwave acoustic properties measuring: phase velocities, quality factor and**

PLS gives us a unique possibility to investigate the acoustic properties within a wide frequency band from several MHz up to tens of GHz: from fundamental λ/2 resonance (*n* = 1) up to UHF

**Figure 7.** General view of the PLS#3 included nine acoustic resonators.

**quality parameter, frequency and temperature dependences**

Microwave studies of PLSs were carried out by equipment (**Figure 6**) comprising the E5071C network analyzer (300 kHz–20 GHz) and the M-150 probe station. The experiments were fulfilled in the reflection mode with a test device connected by ACP40-A-SG-500 probe (the distance between tips was equal to 500 μm). **Figure 7** shows the photograph of one of the

**structures**

176 Piezoelectric Materials

studied samples (PLS #3).

$$Q = \frac{f\_{nr}}{\Delta f} \tag{28}$$

where *fnr* is the resonant frequency of *n*-th overtone of BAW longitudinal type (*L*), and Δ*f* is the bandwidth measured at −3 dB level. The second method for determining the *Q* factor was based on the relation

$$\mathcal{Q}\_{n,\varepsilon\_d} = \frac{1}{2} \alpha \rho\_{p,n} \left| \frac{d\rho\_n}{d\alpha\_{p,n}} \right| = \pi \tau\_d f\_{p,n} \tag{29}$$

where *ω<sup>p</sup>*,*<sup>n</sup>* = 2*πf<sup>p</sup>*,*<sup>n</sup>*, *φ<sup>n</sup>* is the phase shift angle, and *τ<sup>d</sup>* is the group delay time. However, since the experimentally obtained values of *Q* from Eqs. (28) and (29) have only a slight difference, we will further use only Eq. (28) for *Q* calculations.

**Figure 8.** Frequency dependence of *S*11 parameter close to first acoustic overtone (*f*1*<sup>r</sup>* = 8.39543 MHz, and *Q* = 28754) of the PLS #1.

The resonant frequencies of PLS can be calculated approximately by the relation:

$$f\_{nr} \approx n\frac{\nu\_l}{2\hbar},\tag{30}$$

where *vl* is the longitudinal BAW phase velocity, and *h* is the substrate thickness. Calculation of phase velocity for [100] diamond at a given thickness gives us *vl* = 17412 m/s which is close to known data 17542 m/s, obtained by acoustic echo pulse method [7]. Error of ~0.7 % can be explained by TFPT loading and phase shifts in layers.

Note that the resonance phenomena at low frequencies in PLS will be realized with low TFPT effectiveness.

It was found that such parameter as impedance *Z*11 is more convenient than *S*<sup>11</sup> reflection coefficient for the analyzing of PLS acoustic properties. In order to get a proper measurement of a *Q* factor, one needs to obtain a value for a "pure," or extracted, impedance of PLS [20]. First, we measure an absolute value of total impedance *Z*11 at the resonant frequency. Then, the value of the extracted impedance |*Z*11*<sup>e</sup>*|can be calculated as

$$\text{The sum of the same form on the right-hand side}$$

$$\text{The sum of the left-hand side}$$

$$\text{The sum of the right-hand side}$$

$$\text{The sum of the right-hand side}$$

$$\left| Z\_{11\*} \right| = \left| Z\_{11} \right| - \left| Z\_{11}^{\cdot} \right|, \tag{31}$$

**Figure 9.** Frequency dependence of the extracted impedance |*Z*11*<sup>e</sup>*| close to fifth acoustic overtone (*f*5*<sup>r</sup>* = 319.157971 MHz, and *Q* = 47027) of the PLS #2.

where |*Z*<sup>11</sup> ′ | should be measured in the frequency area away from the acoustic resonance. Only after this procedure, the proper *Q* value should be calculated accurately taking |*Z*11*<sup>e</sup>*| magni‐ tudes on the -3 dB level. As an example, frequency dependence of the extracted impedance | *Z*11*<sup>e</sup>*| close to fifth acoustic overtone of PLS #2 is presented in **Figure 9**.

The resonant frequencies of PLS can be calculated approximately by the relation:

*nr*

of phase velocity for [100] diamond at a given thickness gives us *vl*

the value of the extracted impedance |*Z*11*<sup>e</sup>*|can be calculated as

explained by TFPT loading and phase shifts in layers.

where *vl*

178 Piezoelectric Materials

effectiveness.

MHz, and *Q* = 47027) of the PLS #2.

where |*Z*<sup>11</sup>

, 2 *l*

to known data 17542 m/s, obtained by acoustic echo pulse method [7]. Error of ~0.7 % can be

Note that the resonance phenomena at low frequencies in PLS will be realized with low TFPT

It was found that such parameter as impedance *Z*11 is more convenient than *S*<sup>11</sup> reflection coefficient for the analyzing of PLS acoustic properties. In order to get a proper measurement of a *Q* factor, one needs to obtain a value for a "pure," or extracted, impedance of PLS [20]. First, we measure an absolute value of total impedance *Z*11 at the resonant frequency. Then,

'

**Figure 9.** Frequency dependence of the extracted impedance |*Z*11*<sup>e</sup>*| close to fifth acoustic overtone (*f*5*<sup>r</sup>* = 319.157971

after this procedure, the proper *Q* value should be calculated accurately taking |*Z*11*<sup>e</sup>*| magni‐

′ | should be measured in the frequency area away from the acoustic resonance. Only

is the longitudinal BAW phase velocity, and *h* is the substrate thickness. Calculation

*<sup>v</sup> f n <sup>h</sup>* » (30)

11 11 11 , *Z ZZ <sup>e</sup>* = - (31)

= 17412 m/s which is close

**Figure 10.** Frequency dependence of a number of measured PLS parameters close to 16th acoustic overtone (*f*16*<sup>r</sup>* = 1.019209 GHz, and *Q* = 21921) of the PLS #2.

**Figure 11.** Frequency dependence of the extracted impedance |*Z*11*<sup>e</sup>*| close to 425th acoustic overtone (*f*425*<sup>r</sup>* = 9.520235 GHz and *Q* = 9430) of the 1b resonator PLS #3.

A full view of a number of measured PLS parameters such as *Z*11, *Z*11 Real, *Z*11 Imag, *Ze*, *Ze* Real, *Ze* Imag, *Ze* Phase, *Ze* Delay, and Smith diagram is presented in **Figure 10** close to 16th acoustic overtone of the PLS #2. As one can see, the PLS #2 has a high *Q* = 21921 at ~1 GHz of operational frequency. Note that *Ze* phase parameter is convenient in determining the resonant frequency when phase shift is equal to zero. Smith diagram gives us a full screen of the impedance within the given frequency band. For example, this option is convenient when the matching of a device and a measuring circuit should be executed.

Frequency dependence of the extracted impedance |*Z*11*<sup>e</sup>*| close to 425th acoustic overtone of the 1b resonator (PLS #3) is represented in **Figure 11**. Note that a high *Q* = 9430 was obtained at an operational frequency ~10 GHz.

**Figure 12.** Frequency dependence of the extracted impedance *Ze* delay close to 1137th acoustic overtone (*f*1137*<sup>r</sup>* = 19. 8985629 GHz and *Q* = 5969) of the PLS #4.

Resonant peak at extreme operational frequency combining with high *Q* value was obtained at 1137th acoustic overtone (*f*1137*<sup>r</sup>* = 19. 8985629 GHz and *Q* = 5969) of the PLS #4 (**Figure 12**). Note that operational frequency belongs to *K*-band, and as a result the diamond-based PLS can be realized as acoustoelectronic devices efficient at such frequencies.

Analyzing the data on the frequency dependence of *Q* factor within a wide frequency band (**Figure 13**), a nonmonotonic *Q* is noted, decreasing with respect to a frequency increment. The reason for the appearance of local maximums and minimums will be discussed below (Section 4.2). But it is more convenient to represent experimental data by quality parameter as *Q* × *f* product. The main purpose of such representation is that the phonon–phonon attenuation in the crystals is often represented as *Q* × *f* = *f*(ω) relation in order to define the mechanism of UHF acoustic attenuation. Detailed investigation of the frequency dependence of quality parameter for the nine resonators (PLS #3) was carried out, and the experimental results in part are represented in **Figure 14**. Here, rather high values of *Q* (up to 35000) at low frequencies take place, while increasing frequency results in decreasing *Q* factor with ~10000 at 9–10 GHz. Individual resonators differed from one another by configuration and square. As one can see, the best top electrode configuration in terms of higher *Q* × *f* (and *Q*) seems to be the irregular rectangular form. This result is more sufficient at high frequencies and low top electrode area size.

A full view of a number of measured PLS parameters such as *Z*11, *Z*11 Real, *Z*11 Imag, *Ze*, *Ze* Real, *Ze* Imag, *Ze* Phase, *Ze* Delay, and Smith diagram is presented in **Figure 10** close to 16th acoustic overtone of the PLS #2. As one can see, the PLS #2 has a high *Q* = 21921 at ~1 GHz of operational frequency. Note that *Ze* phase parameter is convenient in determining the resonant frequency when phase shift is equal to zero. Smith diagram gives us a full screen of the impedance within the given frequency band. For example, this option is convenient when the matching of a device

Frequency dependence of the extracted impedance |*Z*11*<sup>e</sup>*| close to 425th acoustic overtone of the 1b resonator (PLS #3) is represented in **Figure 11**. Note that a high *Q* = 9430 was obtained

**Figure 12.** Frequency dependence of the extracted impedance *Ze* delay close to 1137th acoustic overtone

Resonant peak at extreme operational frequency combining with high *Q* value was obtained at 1137th acoustic overtone (*f*1137*<sup>r</sup>* = 19. 8985629 GHz and *Q* = 5969) of the PLS #4 (**Figure 12**). Note that operational frequency belongs to *K*-band, and as a result the diamond-based PLS

Analyzing the data on the frequency dependence of *Q* factor within a wide frequency band (**Figure 13**), a nonmonotonic *Q* is noted, decreasing with respect to a frequency increment. The reason for the appearance of local maximums and minimums will be discussed below (Section 4.2). But it is more convenient to represent experimental data by quality parameter as *Q* × *f* product. The main purpose of such representation is that the phonon–phonon attenuation in the crystals is often represented as *Q* × *f* = *f*(ω) relation in order to define the mechanism of

can be realized as acoustoelectronic devices efficient at such frequencies.

and a measuring circuit should be executed.

at an operational frequency ~10 GHz.

180 Piezoelectric Materials

(*f*1137*<sup>r</sup>* = 19. 8985629 GHz and *Q* = 5969) of the PLS #4.

Analyzing **Figure 14**, an unusual frequency dependence of *Q* × *f* quality parameter was found as far as *Q* × *f* has increased at higher frequencies running up to *Q* × *f* ~ 1014 Hz. Such value for diamond is one of the highest among known materials, but for the first time it was obtained at ~10 GHz operational frequency.

**Figure 13.** Frequency dependence of *Q* factor for 1a, 1b, and 1c resonators (PLS #3, see **Figure 7** and Section 3.1).

**Figure 14.** Frequency dependence of quality parameter *Q* × *f* for a set 1a, 1b, and 1c resonators with different top elec‐ trode area and shape (PLS #3, see **Figure 7** and Section 3.1).

**Figure 15.** Frequency dependence of *Z*11 impedance for 1a resonator of PLS #3 measured close to 4.5 (a) and 7 (b) GHz.

Because the PLS in comparison with conventional piezoelectric resonator is an inhomogeneous acoustical device, a lot of unwished waves can be excited, distorting sometimes a useful signal. As an example here, we will focus only on 1a resonator (PLS #3). The amplitude–frequency characteristic (AFC) for this resonator is represented in **Figure 15**. As one can see, at rather low frequencies the main resonant peak has a number of adjacent so-called spurious peaks, which are placed at slightly higher frequencies. Such spurious peaks have lower amplitudes and are located at a certain distance away from the resonant signal. In this case, they have no influence on the main peak and its *Q* value (**Figure 15a**). However, increasing the resonant frequency can result in a convergence of such peak to a resonant signal, and some distortion of a useful signal takes place (**Figure 15b**).

Study of temperature dependence on resonant frequency for diamond-based PLS is also of great practical importance. The experimental results on the temperature dependence of normalized frequency

$$\frac{\Delta f}{f\_r} = \frac{f\_r(T) - f\_r(20\_r)}{f\_r(20\_r)},\tag{32}$$

where *fr*(20*°C*) is the resonant frequency at *T* = 20*°C* (at room temperature) are represented in **Figure 16** for the PLS #5 for two resonant frequencies close to 1 GHz in a wide temperature range. As one can see, the normalized frequency has monotonic temperature dependence with low slope at low-temperature region. Temperature coefficient of frequency (TCF) can be defined as

$$TCF = \frac{1}{40} \frac{f\_r \begin{pmatrix} 40 & \\ \end{pmatrix} - f\_r \begin{pmatrix} 0 & \\ \end{pmatrix}}{f\_r \begin{pmatrix} 20 & \\ \end{pmatrix}}.\tag{33}$$

**Figure 16.** Temperature dependence of ∆*f*/*fr* for the PLS #5 were measured on resonant frequencies *f*58*<sup>r</sup>* = 1101 and *f*56*<sup>r</sup>* = 1063 MHz associated with 58th and 56th overtones, respectively.

For a number of studied PLSs, the TCFs at different frequencies are represented in **Table 2**. The obtained TCF values were in a range(−4.5to − 6.5) × 10− 6*K*− 1.


**Table 2.** TCF for a number of studied PLSs at different frequencies.

**Figure 15.** Frequency dependence of *Z*11 impedance for 1a resonator of PLS #3 measured close to 4.5 (a) and 7 (b) GHz.

Because the PLS in comparison with conventional piezoelectric resonator is an inhomogeneous acoustical device, a lot of unwished waves can be excited, distorting sometimes a useful signal. As an example here, we will focus only on 1a resonator (PLS #3). The amplitude–frequency characteristic (AFC) for this resonator is represented in **Figure 15**. As one can see, at rather low frequencies the main resonant peak has a number of adjacent so-called spurious peaks, which are placed at slightly higher frequencies. Such spurious peaks have lower amplitudes and are located at a certain distance away from the resonant signal. In this case, they have no influence on the main peak and its *Q* value (**Figure 15a**). However, increasing the resonant frequency can result in a convergence of such peak to a resonant signal, and some distortion of a useful

Study of temperature dependence on resonant frequency for diamond-based PLS is also of great practical importance. The experimental results on the temperature dependence of

> ( ) ( ) ( ) <sup>Δ</sup> <sup>20</sup> , <sup>20</sup> *r r*

where *fr*(20*°C*) is the resonant frequency at *T* = 20*°C* (at room temperature) are represented in **Figure 16** for the PLS #5 for two resonant frequencies close to 1 GHz in a wide temperature range. As one can see, the normalized frequency has monotonic temperature dependence with low slope at low-temperature region. Temperature coefficient of frequency (TCF) can be

> ( ) ( ) ( ) <sup>1</sup> 40 0 . 40 20 *r r r*



*r r f fT f f f*

*f f TCF f* °

signal takes place (**Figure 15b**).

normalized frequency

182 Piezoelectric Materials

defined as

The typical temperature dependence of *Q* factor for the PLS #6 is represented in **Figure 17** for a temperature range −100 to 100C. Note that the quality factor decreases while temperature increases.

**Figure 17.** Quality factor's temperature dependence for the PLS #6 measured at resonant frequency ~596 MHz (71st overtone).

In order to excite the SAW, a special interdigital transducer (IDT) should be formed. Study of SAW propagation was carried out by means of a delay line as the "Pt IDT/AlN/(001) diamond" PLS #8 (**Figure 18**). Here, one can see two identical SAW delay lines based on (100) diamond substrate 4 × 4 mm2 , and including IDTs (period *d*IDT = 10 μm, and aperture 500 μm) on the top of the AlN surface. The thickness of the AlN film (~5.5 μm) was chosen to satisfy the condition of effective SAW excitation on the 500–700 MHz range. Therefore, the SAW of Rayleigh-type was propagated along the [110] direction of diamond and simultaneously within the AlN film. In such PLS, the conditions for a number of Rayleigh-type surface waves of different orders could be realized, for example, *R*0, *R*1, and so on.

**Figure 18.** General view of the PLS #8 (*d*IDT = 10 μm).

**Figure 19.** Amplitude–frequency characteristic of SAW propagating in the PLS #8 along [110] direction of diamond: (а) AFC general view; (b) *R*0 mode; (c) *R*1 mode.

The AFC of the PLS #8 is shown in **Figure 19**. There are two resonant peaks: at 449 MHz (fundamental Rayleigh mode *R*0) and at 594 MHz (Sezawa mode *R*1). Such modes are shown on **Figure 19b** and **c**.

#### **3.4. UHF acoustic attenuation in diamond**

In order to excite the SAW, a special interdigital transducer (IDT) should be formed. Study of SAW propagation was carried out by means of a delay line as the "Pt IDT/AlN/(001) diamond" PLS #8 (**Figure 18**). Here, one can see two identical SAW delay lines based on (100) diamond

of the AlN surface. The thickness of the AlN film (~5.5 μm) was chosen to satisfy the condition of effective SAW excitation on the 500–700 MHz range. Therefore, the SAW of Rayleigh-type was propagated along the [110] direction of diamond and simultaneously within the AlN film. In such PLS, the conditions for a number of Rayleigh-type surface waves of different orders

**Figure 19.** Amplitude–frequency characteristic of SAW propagating in the PLS #8 along [110] direction of diamond: (а)

, and including IDTs (period *d*IDT = 10 μm, and aperture 500 μm) on the top

substrate 4 × 4 mm2

184 Piezoelectric Materials

could be realized, for example, *R*0, *R*1, and so on.

**Figure 18.** General view of the PLS #8 (*d*IDT = 10 μm).

AFC general view; (b) *R*0 mode; (c) *R*1 mode.

The results on the frequency dependence of quality parameter can be explained in terms of acoustic attenuation. There are many different mechanisms of acoustical energy losses in PLS. Careful study shows that the attenuation in a rather thick diamond substrate is much higher than the one for thin electrode and AlN film [10]. The influence of roughness losses of AlN film and diamond can be estimated as [20]:

$$
\alpha\_{\rm R} \approx 2\pi \cdot 8.68 \left( k\_{\rm AlN}^2 \eta\_{\rm AlN}^2 + k\_{\rm diam}^2 \eta\_{\rm film}^2 \right) \text{N}\_{\prime} \tag{34}
$$

where *k*AlN, *k*diam, *η*AlN, and *η*diam are the wave vectors and root-mean-squared roughnesses for AlN and diamond, respectively, and *N* is the number of reflections of an acoustic wave in a sample per 1 s. For studied samples with *η*AlN < 30 nm and *η*diam < 15 nm it can be shown that acoustic attenuation contributions concerned with roughness are estimated as: *α<sup>R</sup>*(AlN) ≈ 0.164dB/(GHz2 ×cm) and *α<sup>R</sup>*(diam) ≈ 0.0035dB/(GHz2 ×cm). An extremely low value of damaged layer arising as a result of mechanical polishing of diamond surface was estimated by Kikuchi lines observation [21] as ~30 nm, which is an extremely low value, and has almost no influence on the full acoustic attenuation.

Fundamental origin of acoustical attenuation in solids is defined by phonon–phonon interac‐ tion, which can be concerned with two different mechanisms. At relatively low frequencies, when *ωτth*≪ 1 (*τth* is the time of thermal relaxation for phonon–phonon interaction) Akhiezer's mechanism should take place [22], while at relatively high frequencies (*ωτth* ≫ 1) it should be replaced by Landau–Rumer's mechanism [23]. For such attenuation mechanisms, the quality parameter and acoustic attenuation have a form adapted for PLSs operating by longitudinal BAW [10,24]:

$$\begin{aligned} \alpha\_{\mu\nu\mu\eta} &= \frac{8.68\pi C\_v T \gamma^2 \tau\_{\alpha}}{Q\_v v\_i^3 \left[1 + \left(2\pi \eta \tau\_{\alpha}\right)^2\right]} f^2 \\ Q \times f &= \frac{Q\_v v\_i^3 \left[1 + \left(2\pi \eta \tau\_{\alpha}\right)^2\right]}{C\_v T \gamma^2 \tau\_{\alpha}} \approx \text{const} \end{aligned} \tag{35}$$
 
$$\begin{aligned} \alpha\_{\mu\nu\mu\eta} &= \frac{8.68\pi^3 \gamma^2 \left(k\_a T\right)^4}{30 Q\_v v\_i^3 h^3} f \\ Q \times f &= \frac{30 Q\_v v\_i^5 h^3}{\pi^4 \gamma^2 \left(k\_a T\right)^4} f \end{aligned} \tag{36}$$

Here, *CV* is the specific heat per volume unit, *γ* is Grüneisen parameter, *kB* and *h* are the Boltzmann and Planck's constants, respectively. As one can see, the main difference between Akhiezer and Landau–Rumer mechanisms is the frequency dependence of attenuation and quality parameter. For example, the *Q* × *f* should have no frequency dependence at the Akhiezer regime, while at Landau–Rumer regime, the *Q* × *f* should be increased proportionally to the frequency. Estimation of acoustic attenuation by Eq. (35) within the frequency range 0.3– 10 GHz gives a much higher value than that due to the contributions of roughness taking place on the AlN and diamond surfaces. Thereby, the most significant influence on the resulted acoustic attenuation arises from phonon–phonon interactions. Experimental results of the frequency dependence of quality parameter for a number of diamond-based PLSs are repre‐ sented in **Figure 20**.

As seen from **Figure 20**, the quality parameter has almost no frequency dependence while *f* < 1 GHz, and after that it has a linear growth, which can be explained as the change of the attenuation mechanism at ~1 GHz from Akhiezer's to Landau–Rumer's. Comparing the experimental results from **Figure 20** with the analytical expressions from Eq. (35), the thermal relaxation time and Grüneisen parameter for longitudinal bulk acoustic wave propagating along the [100] direction of diamond can be estimated as 3.5 × 10-10 s and 0.85, respectively. Note that here we used particular Grüneisen parameter for the longitudinal mode since it can be different for different acoustic modes in crystal. Finally, for IIa diamond single crystal at Akhiezer regime (*f* < 1 GHz), *Q* × *f* ≈ 1.8 × 10<sup>13</sup> Hz and α*ph* − *ph* ≈ 0.9 × *f* 2 dB/(GHz2 ×cm), while for Landau–Rumer regime, *Q* × *f* ≈ 1.8 × 104 × *f* Hz and α*ph* <sup>−</sup> *ph* ≈ 0.9 × *f* dB/(GHz×cm). Frequency about 1 GHz is an extremely low value for the change of phonon–phonon interaction mecha‐ nism: all the known crystals applied in acoustoelectronics have such a change at much higher frequencies (20 GHz and higher). This result means that diamond crystal may be a promising substrate for acoustoelectronic devices when the operating frequency should be higher than the units of GHz. UHF acoustic attenuation in diamond in comparison with other commonly used crystals is represented in **Table 3**.

**Figure 20.** Quality parameter's frequency dependence for diamond-based PLSs.


Akhiezer and Landau–Rumer mechanisms is the frequency dependence of attenuation and quality parameter. For example, the *Q* × *f* should have no frequency dependence at the Akhiezer regime, while at Landau–Rumer regime, the *Q* × *f* should be increased proportionally to the frequency. Estimation of acoustic attenuation by Eq. (35) within the frequency range 0.3– 10 GHz gives a much higher value than that due to the contributions of roughness taking place on the AlN and diamond surfaces. Thereby, the most significant influence on the resulted acoustic attenuation arises from phonon–phonon interactions. Experimental results of the frequency dependence of quality parameter for a number of diamond-based PLSs are repre‐

As seen from **Figure 20**, the quality parameter has almost no frequency dependence while *f* < 1 GHz, and after that it has a linear growth, which can be explained as the change of the attenuation mechanism at ~1 GHz from Akhiezer's to Landau–Rumer's. Comparing the experimental results from **Figure 20** with the analytical expressions from Eq. (35), the thermal relaxation time and Grüneisen parameter for longitudinal bulk acoustic wave propagating along the [100] direction of diamond can be estimated as 3.5 × 10-10 s and 0.85, respectively. Note that here we used particular Grüneisen parameter for the longitudinal mode since it can be different for different acoustic modes in crystal. Finally, for IIa diamond single crystal at

about 1 GHz is an extremely low value for the change of phonon–phonon interaction mecha‐ nism: all the known crystals applied in acoustoelectronics have such a change at much higher frequencies (20 GHz and higher). This result means that diamond crystal may be a promising substrate for acoustoelectronic devices when the operating frequency should be higher than the units of GHz. UHF acoustic attenuation in diamond in comparison with other commonly

2

× *f* Hz and α*ph* <sup>−</sup> *ph* ≈ 0.9 × *f* dB/(GHz×cm). Frequency

dB/(GHz2

×cm), while for

Akhiezer regime (*f* < 1 GHz), *Q* × *f* ≈ 1.8 × 10<sup>13</sup> Hz and α*ph* − *ph* ≈ 0.9 × *f*

**Figure 20.** Quality parameter's frequency dependence for diamond-based PLSs.

Landau–Rumer regime, *Q* × *f* ≈ 1.8 × 104

used crystals is represented in **Table 3**.

sented in **Figure 20**.

186 Piezoelectric Materials

**Table 3.** Microwave acoustic attenuation (dB/cm) in diamond in comparison with other crystals at room temperature.
