#
Electro-Acoustic Properties of Scandium-Doped Aluminum Nitride (Sc_{x}Al_{1-x}N) Material and its Application to Phononic Crystal-Coupled SAW Devices

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

_{x}Al

_{1−x}N material were investigated for scandium (Sc) concentrations x = 0 to 0.375. The electro-acoustic properties are used to investigate the frequency response of the SAW delay line, based on the tilt θ° of the normal c-axis of the w-Sc

_{x}Al

_{1−x}N piezoelectric thin film. We found that the piezoelectric response is improved as the Sc concentration increases, which is consistent with existing works in the literature. A 2D-phononic crystal pillars was then grafted on top of the surface, and the dependence of the acoustic band gaps is investigated with the help of the finite element method as a function of the Sc concentration and the tilted angle of w-Sc

_{0.375}Al

_{0.625}N. It was found that the two first band gaps exhibit a shift toward low frequencies with increasing Sc concentration. Moreover, the second acoustic bandgap is more sensitive to the inclination angle than the first. Furthermore, the insertion loss (S

_{21}) of w-Sc

_{0.375}Al

_{0.625}N is improved by 22 dB at θ° = 60°. The c-axis tilted Sc

_{0.375}Al

_{0.625}N-SAW delay line coupled with 2D-phononic crystals is a promising structure for low-loss and high-frequency SAW devices.

## 1. Introduction

^{3+}and Sc

^{3+}for coordination with nitrogen is introduced, which weakens the resistance to nitrogen displacement in the crystal structure and increases the volume of the unit cell. Due to this remarkable piezoelectricity property, scandium (Sc)-doped AlN (ScAlN) is extensively used in high-frequency filters, sensors, and micro-electromechanical devices [15,16,17,18]. Basically, the high piezoelectric constant of ScAlN leads to a significant increase in the electro-mechanical coupling factor, thus resulting in a significant improvement in the performance of SAW devices. Sc

_{x}Al

_{1-x}N alloy thin films may represent an alternative material to replace AlN.

_{33}increases from 6 pC/N for pure AlN to 27.6 pC/N for Sc

_{0.43}Al

_{0.57}N, i.e., at least 500% larger than for AlN. They showed that the piezoelectric response is strongly dependent on the growth temperature. Furthermore, W. Gunilla et al. [21] showed that the electromechanical coupling (k

_{t}

^{2}), influenced by the Sc concentration, increases from 7% in AlN to 15% in Al

_{0.7}Sc

_{0.3}N. As an example, Konno Akira et al. [22] reported that, when using a 40% Sc-doped AlN film, the piezoelectric characteristic of Lamb wave resonators is approximately five times higher than in pure AlN.

_{x}Al

_{1-x}N thin films have been deposited on several substrates such as Si [23], 6H-SiC [24], diamond [25], and sapphire [26]. In each case, the structures enhance the SAW velocity and the electro-mechanical coupling factor k

^{2}. Some examples of SAW devices properties based on Sc

_{x}Al

_{1-x}N/sapphire structures have been investigated using experiments [27] and theories [13,14,28], showing relatively high insertion losses. By comparison, Sc

_{x}Al

_{1-x}N/sapphire structures are promising SAW devices for high-temperature sensor applications [16,29,30].

^{2}), in a tilted ScAlN film on the R-sapphire, is 3.9% at a tilt angle of 90° and 3.7% at a tilt angle of 54° [32]. By comparison, the insertion loss of fabricated IDT/ScAlN tilted at 33°/R-sapphire structure is 34.4 dB, demonstrating that higher electromechanical coupling factor k

^{2}improves the crystal orientation of ScAlN films. The c-axis-tilted ScAlN films with a Sc concentration of 40%, prepared on a silicon substrate via RF magnetron sputtering based on the self-shadowing effect, reach a maximum c-axis tilt angle of 57.4°. In this structure, the electromechanical coupling coefficient (k

^{2}) has been increased because of the c-axis tilt angle [33,34].

_{x}Al

_{1-x}N or c-tilted Sc

_{x}Al

_{1-x}N on PnCs band gaps. For this purpose, because of the dependence of the elastic properties due to scandium concentration and c-axis tilted angle, we first studied the influence of Sc concentration on elastic, piezoelectric, and dielectric properties using the Density Functional Theory (DFT). Then, we used the calculated values of the elastic parameters to investigate the influence of Sc concentration and c-axis tilt angle of Sc

_{x}Al

_{1-x}N thin films on both the acoustic band gap generated by the pillars structuration, and the insertion loss of the SAW device, i.e., the S

_{21}(dB)-scattering parameter.

## 2. Computational Methods

_{x}Al

_{1-x}N Wurtzite crystals. A Monkhorst–Pack mesh of 5 × 5 × 5 k-points in the Brillouin-zone integral was used with a cutoff energy of 500 eV and total energy convergence threshold of 10

^{−6}eV [60]. The Density Functional Theory Perturbation (DFTP) [61] was used to determine both piezoelectric and dielectric constants of w-Sc

_{x}Al

_{1-x}N in the range of x = 0, 0.125, 0.25, and 0.375, and then, the elastic constants (C

_{ij}), the piezoelectric coefficients (e

_{ij}), and the permittivity tensor (ε

_{ij}) of w-Sc

_{x}Al

_{1-x}N. The physical parameters were introduced in the Finite Element model (COMSOL) to obtain the acoustic band gaps of the 2D Sc

_{x}Al

_{1-x}N pillar-based phononic crystals (PnCs), the mechanical transmission, and the transmission loss S

_{21}(dB) of the SAW delay line. The results were compared with the non-structured surface. The same methodology was used in the case of the c-axis tilt angle of w-Sc

_{x}Al

_{1-x}N [62] (see diagram in Figure 1).

## 3. Results and Discussion

_{x}Al

_{1−x}N). For this study, we substituted the concentrations of two, four, and six Al by the Sc atoms in a supercell of (2 × 2 × 1) constructed from a w-AlN Wurtzite structure containing 16 atoms, then representing 12.5%, 25%, and 37.5%. In all concentrations (x = 0, 0.125, 0.25, and 0.375), the predicted lattice parameters of w-Sc

_{x}Al

_{1-x}N agree with the experimental [9] and theoretical results [65] (see Table 1).

_{4}tetrahedron, which deforms when Al atoms are replaced by Sc atoms [19,65]. This effect can then be estimated geometrically from the determination of the lattice constants a and c [9]. Theoretical and experimental details of the impact of scandium concentration on bond length can be found in several papers [9,19,65,66]. Again, these results agree with the experimental [19] and theoretical data [65] obtained from the Quantum Expresso (QE) software package with a (3 × 3 × 1) super cell.

_{x}Al

_{1−x}N obtained from the proposed functional GGA-PBE for various concentrations x = 0, 0.125, 0.25, 0.375, theoretically [64,65,67,68] and experimentally [69,70,71]. Our result agrees with most of the accessible data, indicating the accuracy of our calculation. The elastic stiffness constants Cij should respect the mechanical stability criteria for hexagonal symmetry, as described below [72]:

_{11}−C

_{12}) > 0 and C

_{44}> 0 (C

_{11}+ C

_{12}) C

_{33}−2C

^{2}

_{13}> 0

_{x}Al

_{1-x}N satisfy the above criteria in all compositions over the considered range, indicating that the compounds are mechanically stable. From Figure 2a, we observe that, when the Sc concentration increases, C

_{33}and C

_{11}present an almost linear variation, decreasing from 354 to 220 GPa and from 376 to 282 GPa, respectively, corresponding to a relative decrease of 38 and 25%. Similarly, C

_{44}decreases as a function of the Sc concentration with a relative reduction of 25%. By comparison, doping AlN with Sc increases the mixed compression/shear C

_{12}and C

_{13}elastic constants. This behavior is similar with the observed theoretical [66,73,74] and experimental [9] works. We can therefore conclude that, as the concentration of Sc increases, the material softens along the c-axis, and hardens in the basal plane.

#### 3.1. Piezoelectric and Dielectric Constants

_{ik}and the elastic constants Cij as follows [75]:

_{ij}is given by:

_{x}Al

_{1-x}N, which are the sum of the ionic and electronic contributions. Table 2 reports our calculated values, which are in good agreement with other theoretical data and measured values.

_{15}, e

_{31}, e

_{33}) as a function of the Sc concentration. The most striking feature of this graph is the quick evolution of e

_{33}, which increases by 67%. By comparison, the two coefficients e

_{15}and e

_{31}show a relative increase of 18 and 2.75%, respectively. This result indicates that the addition of a trivalent dopant (Sc

^{3+}) constrains the displacement along the c-axis. Moreover, because the electronegativity of the Sc atom is lower than that of the Al atom, the w-Sc

_{x}Al

_{1-x}N material is more electrovalent, which improves the piezoelectric properties [19].

_{15}, e

_{31}, and e

_{33}with increasing Sc content is consistent and follows the general trend obtained from previous theoretical results [64,65,67,68] and experimental measurements [9,69,70].

_{11}and ε

_{33}at x = 0% are in good agreement with previous data [68,69,70]. Then, as seen in Figure 3, the two components ε

_{33}and ε

_{11}increase almost linearly with the concentration of Sc [21]. Obviously, ε

_{33}is more sensitive than ε

_{11}, increasing by 38% compared to 26%. This is mainly due to the strong lattice polarization induced by the out-of-plane (c-axis) Sc atom.

_{x}Al

_{1-x}N. All these results show an enhanced electromechanical coupling (k

^{2}), which justify the selection of w-Sc

_{0.375}Al

_{0.625}N material for SAW devices.

#### 3.2. Dependence on Electro-Acoustic Parameters of w-Sc_{0.375}Al_{0.625}N with tilted c-Axis Orientation

_{0.375}Al

_{0.625}N, a rotation following a clockwise angle θ against the y-axis was considered. The original coordinates (x, y, z) were changed to a set of new coordinates (x

^{′}, y

^{′}, z

^{′}). It was found that the elastic stiffness C

_{ij}, the piezoelectric stress e

_{ij}coefficient, and the dielectric permittivity

**ε**

_{ij}can be computed through properties in original coordinate system (x

^{′}, y

^{′}, z

^{′}) with the help of matrix algebra [76].

_{ij}against the inclination angle θ. It is clearly seen that, when changing θ from −90° to 90

^{°}, a symmetric behavior for all C

_{ij}is observed at θ = 0°. The increase in θ leads to decreases in C

_{11}, C

_{12}, and C

_{13}. An opposite dependence is observed between C

_{33}and C

_{11}due to the c-axis inclination angle, as these two elements are related to the out-of-plane and in-plane component. It should be mentioned that C

_{44}does not depend on the inclination angle, which can be explained by the fact that x

_{2}and x

_{2}

^{′}present the same axis.

_{ij}with the c-axis tilted angle. We observed that e

_{15}and e

_{31}are negative at 0° and behave as a sinus. With the increase of the inclination angle θ, e

_{15}and e

_{31}reach a maximum at 56° and 58° respectively. This dependence was also observed in c-axis-tilted AlN with the same trends [78]. In addition, e

_{33}reaches a maximum at θ = 0°, a minimum at θ = 70°, then becomes zero at 90°, in addition to e

_{15}and e

_{33}.

_{11}and ε

_{33}as a function of the c-axis tilted angle θ

^{°}. The dielectric constants ε

_{11}and ε

_{33}present an opposite trend with equal values at (θ = 45°). Indeed, the absolute value of ε

_{11}is maximum at θ = 90°, while ε

_{33}component has a maximum at θ = 0°, equivalent to the behavior of undoped AlN [78]. This is attributed to the fact that the x

_{1}

^{′}axis becomes x

_{3}, and x

_{3}

^{′}coincides with negative x

_{1}when θ is equal to 90°, and is a direct consequence of the Wurtzite crystal structure of group III-Nitrides [79,80].

#### 3.3. FEM Simulation of PnCs Unit Cell Dispersion Modes

_{2}O

_{3}substrate, arranged in a square lattice. The elementary unit cell, shown in Figure 5a, is repeated periodically in the (x, y) plane with the pillar axis oriented along the z-axis. The filling factor is defined by f = π.r

^{2}/a

^{2}, where a is the lattice parameter of the phononic crystal and r the radius of the cylindrical pillar. The pillar height is h

_{1}, h

_{2}is the thickness of the AlN layer, and h

_{3}is the thickness of the bulk material (Al

_{2}O

_{3}), chosen to be five times the lattice constant of the PnCs unit cell (h

_{3}= 5 × a) [81].

_{B}) (in the x and y directions) are swept between the high symmetry points of the first irreducible Brillouin zone (Γ-X-M-Γ) represented Figure 5b.

#### 3.3.1. Effect of the Structural Parameters

_{c}and the bandgap width B

_{w}. These two indicators determine the operating parameters of the PnCs and are defined as [55]:

_{u}and f

_{l}represent respectively the upper and lower frequency limits of the band gap. The band gaps operating parameters have been investigated as a function of the geometrical parameters, namely the pillar’s height (h) and radius (r). Figure 7a–c show the phononic crystal band structure for three different characteristic values of (h), i.e., 4, 6, and 8 µm, with a fixed radius value of r = 3 µm and a = 8 µm. The dependence of the forbidden band according to the height of the pillars is highlighted. The absolute band gap of the SAWs is limited to the domain below the sound line. When h = 4 µm, only one absolute band gap is observed (Figure 7a), whereas two band gaps occur for 6 µm, then three for 8 µm. Additionally, when the height of the pillars h increases, the band gaps downshift toward low frequencies and new band gaps appear. According to [58,83], they can be attributed to local resonance bandgaps.

_{c}and B

_{w}of the first and second bandgap as a function of h, keeping constant a = 8 µm and r = 2.6 µm. For the first (resp. second) band gap, the width B

_{w}decreases from 98 MHz (resp. 45 MHz) to 78 MHz (resp. 24 MHz) when h increases from 4 µm (resp. 6µm) to h = 8 µm. Furthermore, as shown in Figure 8b, the center frequency f

_{c1}decreases almost linearly from 339 to 121 MHz when h changes from 2.4 to 8 µm, and f

_{c2}decreases from 331 to 252 MHz when h changes from 5.6 to 8 µm.

#### 3.3.2. Effect of AlN Doping on the Acoustic Band Gaps

_{c}and the bandwidth B

_{w}. Indeed, as seen Figure 9a, when the Sc concentration increases, both first and second band gaps exhibit a consequent shift down toward low frequencies, while their widths are slightly modified. This is confirmed by a representation of f

_{c}and B

_{w}as a function of the Sc concentration (Figure 9b). From this figure, we notice that the center frequencies vary linearly from 161 to 131 MHz and from 321 to 259 MHz when the Sc concentration increases from 0 to 37.5% for the first and second acoustic band gaps, respectively. Such a variation is due to the acoustic velocity which is modified by the elastic constant C

_{ij}and density ρ (g/cm

^{3}) of w-Sc

_{x}Al

_{1-x}N. For the two band gaps, the bandgap width (B

_{w}) has been slightly modified, with a variation of −10% for the first gap and +7% for the second.

_{0.375}Al

_{0.625}N PnC-based structure. From the results shown in Figure 10a,b, it is clear that when θ changes from −90° to 90°, a symmetric behavior with respect to θ = 0° is observed for the two acoustic bandgaps, due to the dependence of C

_{ij}on the angle of inclination. Additionally, we can see that the second gap is more sensitive to c-axis inclination angle (see Figure 10b). All the modification of the physical parameters can provide a new perspective for controlling SAW-coupled PnC devices and applications.

#### 3.3.3. Analysis of Surface Phononic Modes in the SAW-PnCs

_{0.375}Al

_{0.625}N pillars with a = 8 µm, h = 6 µm, and r = 2.6 µm, and in Figure 11b the corresponding mechanical displacement field of the first modes, designated by points A, B, C, D, and E. The acoustic branches, passing by A and D, are degenerated because of the x and y polarization modes appearing at the same frequency.

_{0.375}Al

_{0.625}N/Al

_{2}O

_{3}structure).

_{out}/u

_{in}), was then recorded after the phononic crystal (Figure 11b). The transmission spectrum presents zero transmissions at frequencies 88, 205, and 282 MHz, which correspond to the frequencies of modes A, C, and D, respectively. One can note that mode C is at the origin of a large band gap compared to modes A and D, while mode B does not lead to any effect in the transmission curve. This is due to the symmetry of mode B, which cannot be excited under the SAW.

_{0.625}Al

_{0.375}N phononic crystal was set between the IDTs transmitter and receiver (see Figure 12a). To avoid time-consuming calculations and to focus on the physical effects induced by the Sc concentration, we performed all the following calculations based on a 2D model. In that case, the pillared structure will be transformed in a phononic crystal made of ridges, oriented infinitely along y. Compared to the 3D pillared crystal, it results eigenmodes A′, B′, and D′ operating now at the corresponding frequencies 89, 190, and 277 MHz, obtained by the calculation of dispersion curves and transmission curves performed on the 2D model (see Supplementary Materials).

_{in}= 1 V (input), while the odd ones are connected to the ground. In the output, V

_{out}is connected between the even electrodes and the odd ones as ground.

_{x}Al

_{1-x}N piezoelectric layer of 2 µm thickness in the x direction. Figure 12b shows the insertion loss (S

_{21}) for different values of Sc concentration, i.e., x = 0, 12.5, 25, and 37.5%. As seen in Figure 12b, we were able to generate an acoustic surface pulse in the frequency range (355 MHz, 385 MHz) for pure AlN (x = 0%). When now increasing the Sc concentration, the maximum of the transmitted amplitude increases and shifts toward the low frequencies, from 371 MHz (x = 0%) to 360 MHz (x = 37.5%). Under the same conditions of excitation $\left(\lambda =16\mu m\right)$ and for pure AlN (x = 0%), the deviation in the center frequency of the SAW device from the experimental value of 355 MHz obtained by Ginlinger et al. [27] is 4.5%. Moreover, their experimental work shows that doping the AlN (27% Sc) enhanced the performance of Sc

_{x}Al

_{1-x}N-based SAW devices.

_{21}) of Sc

_{x}Al

_{1-x}N-based delay line has been increased from −34.52, −32.56, −30.56, and −28.81 dB for the respective concentrations x = 0, 12.5, 25, and 37.5%. This behavior agrees the conclusion that the insertion loss (S

_{21}) of pure AlN can be improved by a Sc concentration of 37.5%.

_{21}of the Al/Sc

_{x}Al

_{1-x}N/Sapphire for x = 37.5%, obtained with and without the phononic crystal at different wavelengths. The wavelengths were chosen in order to track the frequencies’ eigenmodes of the ridges, namely A′, C′, and D′. To define the appropriated wavelengths, we used the expression f = v

_{saw}/λ, where the surface wave velocity results from the calculation at x = 37.5%. We found that the wavelengths λ = 64 μm, λ = 30 μm, and λ = 20 μm cover respectively the eigenfrequencies A′, C′, and D′.

_{21}are affected in the vicinity of the SAW central frequency. In other words, the propagation of the SAW is disturbed by the presence of the ridged structure, resulting in a decrease in the mechanical energy of the IDT receiver. S

_{21}also presents a slight frequency shift toward low frequencies because of the mass loading effect caused by the presence of the PnC [87,88]. S

_{21}in the delay line also shows small oscillations, which are induced by Fabry–Perot reflection from the metallic IDT fingers [89].

_{ij}(Figure 4a), and the piezoelectric e

_{ij}(Figure 4b) and dielectric values ɛ

_{ij}(Figure 4c), the dependence of the insertion loss (S

_{21}) of the SAW delay line without PnCs versus the c-axis tilted angle (θ°) of Sc

_{0.375}Al

_{0.625}N was examined for θ = 20°, 40°, 60° and 80°, and compared to that when the c-axis is normally oriented (θ = 0°). The result is reported in Figure 14a, which shows that the insertion loss (S

_{21}) is affected by the c-axis tilting. The variation in the insertion loss is directly linked to the electromechanical coupling coefficient, which is very sensitive to the tilted angle [90].

_{15}and e

_{31}of Sc

_{0.375}Al

_{0.625}N thin films (see Figure 4b). In addition, the variation om the resonances frequencies as a function of x behaves as the calculated SAW velocities at λ= 20 µm (5531.4, 5473.2, 5495.6, 5466.4, and 5405 m/s).

_{0.375}Al

_{0.625}N SAW delay line compared to the pure AlN SAW delay line. The same improvement was observed experimentally by A. Kochar et al. [26,30]. Moreover, the passing frequency band of the SAW delay line based on 60° inclined pure AlN is shifted down to the low frequencies compared to Sc

_{0.375}Al

_{0.625}N at the same angle. This is due to the influence of the Sc concentration on the acoustic velocity [91].

## 4. Conclusions

_{x}Al

_{1-x}N in the x range from 0 to 37.5% by means of Density Functional Theory. The calculated elastic, piezoelectric, and dielectric properties using the GGA-PBE function show very good agreement with experiments and theoretical works. For all considered Sc contents, w-Sc

_{x}Al

_{1-x}N material exhibits good mechanical stability criteria. By increasing the Sc concentrations for pure AlN, the elastic constants C

_{11}and C

_{33}decrease, whereas the piezoelectric (e

_{33}) and dielectric (ε

_{33}) constants increase, enhancing the performance of the SAW devices based on w-Sc

_{0.375}Al

_{0.625}N.

_{ij}, e

_{ij}, and ε

_{ij}) with the tilted angle of w-Sc

_{0.375}Al

_{0.625}N was investigated. It is observed that all these properties exhibit a symmetric behavior at a 0° tilted angle, whereas an opposite trend for both C

_{11,}C

_{33,}ε

_{11,}and ε

_{33}was found at the tilted angle of 45°. The material becomes non-piezoelectric at the tilted angle of 90°.

_{0.375}Al

_{0.625}N calculated by DFT were used to calculate the dispersion curves of a pillared phononic crystal deposited on top of the substrate. The effects of scandium (Sc) concentration and tilted angle θ

^{°}of w-Sc

_{0.375}Al

_{0.625}N on acoustic band gaps and S

_{21}scattering parameters were studied for the first time. The geometrical, (h = 6 μm and r = 2.6 µm), and physical (x = 37.5%) parameters were found to be the appropriate choice to obtain a maximum bandwidth of 44.83 MHz. By comparison, a symmetric behavior at a 0° tilted angle was also revealed for larger acoustic band gaps. An improvement in the S

_{21}intensity (in dB) of SAW delay lines is demonstrated when x = 37.5% for a normally oriented c-axis. We found that 60

^{°}is an optimal tilted angle to improve the insertion loss (S

_{21}) from −19 dB for AlN to −12.8 dB for w-Sc

_{0.375}Al

_{0.625}N. Ongoing work is dealing with AlScN-SAW devices coupled with 2D phononic crystal as a highly sensitive micro-sensor for liquid property determination.

## Supplementary Materials

_{c}and the widths B

_{w}of the first and second bandgap as a function of r. The figure S3 shows the dispersion curves, the transmission spectrum and displacement field of modes A’, B’, and C’.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Panneerselvam, G.; Thirumal, V.; Pandya, H.M. Review of surface acoustic wave sensors for the detection and identification of toxic environmental Gases/Vapours. Arch. Acoust.
**2019**, 44, 117–127. [Google Scholar] - Caliendo, C.; Imperatori, P. High-frequency, high-sensitivity acoustic sensor implemented on ALN/Si substrate. Appl. Phys. Lett.
**2003**, 83, 1641–1643. [Google Scholar] [CrossRef] - Korovin, A.V.; Pennec, Y.; Stocchi, M.; Mencarelli, D.; Pierantoni, L.; Makkonen, T.; Ahopelto, J.; Rouhani, B.D. Conversion between surface acoustic waves and guided modes of a quasi-periodic structured nanobeam. J. Phys. D Appl. Phys.
**2019**, 52, 32LT01. [Google Scholar] [CrossRef] - Water, W.; Yan, Y.-S.; Meen, T.-H. Effect of magnesium doping on the structural and piezoelectric properties of sputtered ZnO thin film. Sens. Actuators A: Phys.
**2008**, 144, 105–108. [Google Scholar] [CrossRef] - Toyama, M.; Kubo, R.; Takata, E.; Tanaka, K.; Ohwada, K. Characterization of piezoelectric properties of PZT thin films deposited on Si by ECR sputtering. Sens. Actuators A: Phys.
**1994**, 45, 125–129. [Google Scholar] [CrossRef] - Lin, C.-M.; Chen, Y.-Y.; Felmetsger, V.V.; Lien, W.-C.; Riekkinen, T.; Senesky, D.; Pisano, A.P. Surface acoustic wave devices on AlN/3C–SiC/Si multilayer structures. J. Micromech. Microeng.
**2013**, 23, 25019. [Google Scholar] [CrossRef] - Deger, C.; Born, E.; Angerer, H.; Ambacher, O.; Stutzmann, M.; Hornsteiner, J.; Riha, E.; Fischerauer, G. Sound velocity of AlxGa1−xN thin films obtained by surface acoustic-wave measurements. Appl. Phys. Lett.
**1998**, 72, 2400–2402. [Google Scholar] [CrossRef] - Doll, J.C.; Petzold, B.C.; Ninan, B.; Mullapudi, R.; Pruitt, B.L. Aluminum nitride on titanium for CMOS compatible piezoelectric transducers. J. Micromechanics Microengineering
**2009**, 20, 25008. [Google Scholar] [CrossRef] [Green Version] - Kurz, N.; Ding, A.; Urban, D.F.; Lu, Y.; Kirste, L.; Feil, N.M.; Žukauskaitė, A.; Ambacher, O. Experimental determination of the electro-acoustic properties of thin film AlScN using surface acoustic wave resonators. J. Appl. Phys.
**2019**, 126, 075106. [Google Scholar] [CrossRef] - Mayrhofer, P.; Riedl, H.; Euchner, H.; Stöger-Pollach, M.; Bittner, A.; Schmid, U. Microstructure and piezoelectric response of Y Al1−N thin films. Acta Mater.
**2015**, 100, 81–89. [Google Scholar] [CrossRef] [Green Version] - Yanagitani, T.; Jia, J. ScAlN polarization inverted resonators and enhancement of k
_{t}^{2}in new YbAlN materials for BAW devices. In Proceedings of the IEEE International Ultrasonics Symposium (IUS); 2019; pp. 894–899. [Google Scholar] [CrossRef] - Assali, A.; Laidoudi, F.; Serhane, R.; Kanouni, F.; Mezilet, O. Highly Enhanced Electro-acoustic Properties of YAlN/Sapphire Based Surface Acoustic Wave Devices for Next Generation of Microelectromechanical Systems. Mater. Today Commun.
**2021**, 26, 102067. [Google Scholar] [CrossRef] - Laidoudi, F.; Amara, S.; Caliendo, C.; Boubenider, F.; Kanouni, F.; Assali, A. High quality and low loss surface acoustic wave SAW resonator based on chromium-doped AlN on sapphire. Appl. Phys. A
**2021**, 127, 1–11. [Google Scholar] [CrossRef] - Amara, S.; Kanouni, F.; Laidoudi, F.; Bouamama, K. Low loss surface acoustic wave SAW devices based on Al1-xMxN (M=Cr, Y, Sc) thin films. Phys. B: Condens. Matter
**2021**, 615, 412990. [Google Scholar] [CrossRef] - Fei, C.; Liu, X.; Zhu, B.; Li, D.; Yang, X.; Yang, Y.; Zhou, Q. AlN piezoelectric thin films for energy harvesting and acoustic devices. Nano Energy
**2018**, 51, 146–161. [Google Scholar] [CrossRef] - Aubert, T.; Naumenko, N.; Bartoli, F.; Pigeat, P.; Streque, J.; Ghanbaja, J.; Elmazria, O. Non-leaky longitudinal acoustic modes in ScxAl1-xN/sapphire structure for high-temperature sensor applications. Appl. Phys. Lett.
**2019**, 115, 83502. [Google Scholar] [CrossRef] - Qi, W.; Lu, Y.; Fung, S.; Jiang, X.; Horsley, D. Scandium Doped Aluminum Nitride Based Piezoelectric Micromachined Ultrasound Transducers. In Proceedings of the Hilton Head Workshop 2016: A Solid-State Sensors, Actuators and Microsystems Workshop, Hilton Head, SC, USA, 5–9 June 2016. [Google Scholar]
- Casamento, J.; Chang, C.S.; Shao, Y.-T.; Wright, J.; Muller, D.A.; Xing, H.; Jena, D. Structural and piezoelectric properties of ultra-thin Sc
_{x}Al_{1−x}N films grown on GaN by molecular beam epitaxy. Appl. Phys. Lett.**2020**, 117, 112101. [Google Scholar] [CrossRef] - Akiyama, M.; Kamohara, T.; Kano, K.; Teshigahara, A.; Takeuchi, Y.; Kawahara, N. Enhancement of Piezoelectric Response in Scandium Aluminum Nitride Alloy Thin Films Prepared by Dual Reactive Cosputtering. Adv. Mater.
**2008**, 21, 593–596. [Google Scholar] [CrossRef] - Akiyama, M.; Kano, K.; Teshigahara, A. Influence of growth temperature and scandium concentration on piezoelectric response of scandium aluminum nitride alloy thin films. Appl. Phys. Lett.
**2009**, 95, 162107. [Google Scholar] [CrossRef] - Wingqvist, G.; Tasnadi, F.; Zukauskaite, A.; Birch, J.; Arwin, H.; Hultman, L. Increased electromechanical coupling in w−ScxAl1−xN. Appl. Phys. Lett.
**2010**, 97, 112902. [Google Scholar] [CrossRef] - Konno, A.; Sumisaka, M.; Teshigahara, A.; Kano, K.; Hashimo, K.-Y.; Hirano, H.; Esashi, M.; Kadota, M.; Tanaka, S.; Konno, A.; et al. ScAlN Lamb wave resonator in GHz range released by XeF2 etching. In Proceedings of the 2013 IEEE International Ultrasonics Symposium (IUS), Prague, Czech Republic, 21–25 July 2013; 2013; pp. 1378–1381. [Google Scholar] [CrossRef]
- Wang, W.; Mayrhofer, P.M.; He, X.; Gillinger, M.; Ye, Z.; Wang, X.; Bittner, A.; Schmid, U.; Luo, J. High performance AlScN thin film based surface acoustic wave devices with large electromechanical coupling coefficient. Appl. Phys. Lett.
**2014**, 105, 133502. [Google Scholar] [CrossRef] - Teshigahara, A.; Hashimoto, K.-Y.; Akiyama, M. Scandium aluminum nitride: Highly piezoelectric thin film for RF SAW devices in multi GHz range. In Proceedings of the IEEE International Ultrasonics Symposium, Dresden, Germany, 7–10 October 2012; pp. 1–5. [Google Scholar] [CrossRef]
- Wu, S.; Wu, M.Y.; Huang, J.-L.; Lii, D.-F. Characterization and Piezoelectric Properties of Reactively Sputtered (Sc, Al)N Thin Films on Diamond Structure. Int. J. Appl. Ceram. Technol.
**2013**, 11, 894–900. [Google Scholar] [CrossRef] - Kochhar, A.; Yamamoto, Y.; Teshigahara, A.; Hashimoto, K.-Y.; Tanaka, S.; Esashi, M. Wave Propagation Direction and c-Axis Tilt Angle Influence on the Performance of ScAlN/Sapphire-Based SAW Devices. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2016**, 63, 953–960. [Google Scholar] [CrossRef] - Gillinger, M.; Shaposhnikov, K.; Knobloch, T.; Schneider, M.; Kaltenbacher, M.; Schmid, U. Impact of layer and substrate properties on the surface acoustic wave velocity in scandium doped aluminum nitride based SAW devices on sapphire. Appl. Phys. Lett.
**2016**, 108, 231601. [Google Scholar] [CrossRef] - Kanouni, F.; Amara, S.; Assali, A.; Arab, F.; Qin, Z. A P-matrix-based model for the frequency analysis of IDT/AlScN/Sapphire SAW-delay line. Sens. Actuators A Phys.
**2020**, 307, 111980. [Google Scholar] [CrossRef] - Gillinger, M.; Knobloch, T.; Schneider, M.; Schmid, U. Harsh Environmental Surface Acoustic Wave Temperature Sensor Based on Pure and Scandium doped Aluminum Nitride on Sapphire. Multidiscip. Digit. Publ. Inst. Proc.
**2017**, 1, 341. [Google Scholar] [CrossRef] [Green Version] - Bartoli, F.; Moutaouekkil, M.; Streque, J.; Pigeat, P.; Hage-Ali, S.; Boulet, P.; M’Jahed, H.; Elmazria, O.; Zhgoon, S.; Aubert, T.; et al. Theoretical and experimental study of ScAlN/Sapphire structure based SAW sensor. IEEE Sens.
**2017**, 1–3. [Google Scholar] [CrossRef] [Green Version] - Kochhar, A.; Yamamoto, Y.; Teshigahara, A.; Hashimoto, K.-Y.; Tanaka, S.; Esashi, M. NSPUDT using c-axis tilted ScAlN thin film. In Proceedings of the Joint Conference of the IEEE International Frequency Control Symposium & the European Frequency and Time Forum, Orlando, FL, USA, 14–18 April 2015; pp. 633–636. [Google Scholar] [CrossRef]
- Tokuda, S.; Takayanagi, S.; Matsukawa, M.; Yanagitani, T. Film growth of c-axis tilted ScAlN on the sapphire substrate for SAW devices. In Proceedings of the 2017 IEEE International Ultrasonics Symposium (IUS), Washington, DC, USA, 6–9 September 2017; pp. 1–4. [Google Scholar] [CrossRef]
- Tominaga, T.; Takayanagi, S.; Yanagitani, T. c-axis-tilted ScAlN film on silicon substrate for surface acoustic wave device. In Proceedings of the 2021 IEEE International Ultrasonics Symposium (IUS), Xi’an, China, 11–16 September 2021; pp. 1–4. [Google Scholar] [CrossRef]
- Tominaga, T.; Takayanagi, S.; Yanagitani, T. c-Axis-tilted ScAlN films grown on silicon substrates for surface acoustic wave devices. Jpn. J. Appl. Phys.
**2022**, 61, SG1054. [Google Scholar] [CrossRef] - Djafari-Rouhani, B.; Maradudin, A.A.; Wallis, R.F. Rayleigh waves on a superlattice stratified normal to the surface. Phys. Rev. B
**1984**, 29, 6454–6462. [Google Scholar] [CrossRef] - Tanaka, Y.; Tamura, S.-I. Surface acoustic waves in two-dimensional periodic elastic structures. Phys. Rev. B
**1998**, 58, 7958–7965. [Google Scholar] [CrossRef] [Green Version] - Wu, T.-T.; Wu, L.-C.; Huang, Z.-G. Frequency band-gap measurement of two-dimensional air/silicon phononic crystals using layered slanted finger interdigital transducers. J. Appl. Phys.
**2005**, 97, 094916. [Google Scholar] [CrossRef] - Hsu, J.-C.; Lin, Y.-D. Microparticle concentration and separation inside a droplet using phononic-crystal scattered standing surface acoustic waves. Sens. Actuators A Phys.
**2019**, 300, 111651. [Google Scholar] [CrossRef] - Benchabane, S.; Khelif, A.; Rauch, J.-Y.; Robert, L.; Laude, V. Evidence for complete surface wave band gap in a piezoelectric phononic crystal. Phys. Rev. E
**2006**, 73, 65601. [Google Scholar] [CrossRef] [Green Version] - Yudistira, D.; Pennec, Y.; Rouhani, B.D.; Dupont, S.; Laude, V. Non-radiative complete surface acoustic wave bandgap for finite-depth holey phononic crystal in lithium niobate. Appl. Phys. Lett.
**2012**, 100, 61912. [Google Scholar] [CrossRef] [Green Version] - White, R.M.; Voltmer, F.W. Direct piezoelectric Coupling to surface elastic waves. Appl. Phys. Lett.
**1965**, 7, 314–316. [Google Scholar] [CrossRef] - Sigalas, M.; Economou, E. Elastic and acoustic wave band structure. J. Sound Vib.
**1992**, 158, 377–382. [Google Scholar] [CrossRef] - Li, G.; Ma, F.; Guo, J.; Zhao, H. Case Study of Roadway Deformation Failure Mechanisms: Field Investigation and Numerical Simulation. Energies
**2021**, 14, 1032. [Google Scholar] [CrossRef] - Mohammadi, S.; Adibi, A. On chip complex signal processing devices using coupled phononic crystal slab resonators and waveguides. AIP Adv.
**2011**, 1, 41903. [Google Scholar] [CrossRef] - Wu, T.; Sun, J. 4G-3 guided surface acoustic waves in phononic crystal waveguides. In Proceedings of the 2006 IEEE Ultrasonics Symposium 2006, Vancouver, BC, Canada, 3–6 October 2006; pp. 673–676. [Google Scholar]
- Salman, A.; Kaya, O.A.; Cicek, A. Determination of concentration of ethanol in water by a linear waveguide in a 2-dimensional phononic crystal slab. Sens. Actuators A Phys.
**2014**, 208, 50–55. [Google Scholar] [CrossRef] - Wu, T.; Wang, W.; Sun, J. A layered SAW device using phononic-crystal reflective gratings. In Proceedings of the 2008 IEEE Ultrasonics Symposium, Beijing, China, 2–5 November 2008; pp. 709–712. [Google Scholar]
- Imanian, H.; Noori, M.; Abbasiyan, A. Highly efficient gas sensor based on quasi-periodic phononic crystals. Sens. Actuators B Chem.
**2021**, 345, 130418. [Google Scholar] [CrossRef] - Ramakrishnan, N.; Palathinkal, R.P.; Nemade, H.B. Mass Loading Effect of High Aspect Ratio Structures Grown Over Surface Acoustic Wave Resonators. Sens. Lett.
**2010**, 8, 253–257. [Google Scholar] [CrossRef] - Benchabane, S.; Gaiffe, O.; Ulliac, G.; Salut, R.; Achaoui, Y.; Laude, V. Observation of surface-guided waves in holey hypersonic phononic crystal. Appl. Phys. Lett.
**2011**, 98, 171908. [Google Scholar] [CrossRef] - Yankin, S.; Talbi, A.; Du, Y.; Gerbedoen, J.-C.; Preobrazhensky, V.; Pernod, P.; Matar, O.B. Finite element analysis and experimental study of surface acoustic wave propagation through two-dimensional pillar-based surface phononic crystal. J. Appl. Phys.
**2014**, 115, 244508. [Google Scholar] [CrossRef] - Binci, L.; Tu, C.; Zhu, H.; Lee, J.E.-Y. Planar ring-shaped phononic crystal anchoring boundaries for enhancing the quality factor of Lamb mode resonators. Appl. Phys. Lett.
**2016**, 109, 203501. [Google Scholar] [CrossRef] - Ardito, R.; Cremonesi, M.; D’Alessandro, L.; Frangi, A. Application of optimally-shaped phononic crystals to reduce anchor losses of MEMS resonators. In Proceedings of the IEEE International Ultrasonics Symposium (IUS), Montreal, QC, Canada, 5–8 September 2016; pp. 1–3. [Google Scholar] [CrossRef] [Green Version]
- Siddiqi, M.W.U.; Lee, J.E.-Y. Quality factor enhancement of AlN-on-Si lamb wave resonators using a hybrid of phononic crystal shapes in anchoring boundaries. In Proceeding of the 20th International Conference on Solid-State Sensors, Actuators and Microsystems &Eurosensors XXXIII (Transducers & Eurosensors XXXIII), Berlin, Germany, 23–27 June 2019; pp. 913–916. [Google Scholar] [CrossRef]
- Tong, Y.; Han, T. Anchor Loss Reduction of Lamb Wave Resonator by Pillar-Based Phononic Crystal. Micromachines
**2021**, 12, 62. [Google Scholar] [CrossRef] - Pourabolghasem, R.; Dehghannasiri, R.; Eftekhar, A.A.; Adibi, A. Waveguiding Effect in the Gigahertz Frequency Range in Pillar-based Phononic-Crystal Slabs. Phys. Rev. Appl.
**2018**, 9, 14013. [Google Scholar] [CrossRef] [Green Version] - Dehghannasiri, R.; Eftekhar, A.A.; Adibi, A. Hypersonic Surface Phononic Bandgap Demonstration in a CMOS-Compatible Pillar-Based Piezoelectric Structure on Silicon. Phys. Rev. Appl.
**2018**, 10, 64019. [Google Scholar] [CrossRef] - Taleb, F.; Darbari, S. Tunable Locally Resonant Surface-Acoustic-Waveguiding Behavior by Acoustoelectric Interaction in ZnO -Based Phononic Crystal. Phys. Rev. Appl.
**2019**, 11, 24030. [Google Scholar] [CrossRef] - Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett.
**1996**, 77, 3865. [Google Scholar] [CrossRef] [Green Version] - Monkhorst, H.J.; Pack, J.D. Special points for Brillouin-zone integrations. Phys. Rev. B
**1976**, 13, 5188. [Google Scholar] [CrossRef] - Baroni, S.; Giannozzi, P.; Testa, A. Green’s-function approach to linear response in solids. Phys. Rev. Lett.
**1987**, 58, 1861–1864. [Google Scholar] [CrossRef] - COMSOL Multiphysics. Available online: https://www.comsol.com/ (accessed on 10 September 2022).
- Schulz, H.; Thiemann, K. Crystal structure refinement of AlN and GaN. Solid State Commun.
**1977**, 23, 815–819. [Google Scholar] [CrossRef] - Momida, H.; Teshigahara, A.; Oguchi, T. Strong enhancement of piezoelectric constants in Sc
_{x}Al_{1−x}N: First-principles calculations. AIP Adv.**2016**, 6, 65006. [Google Scholar] [CrossRef] [Green Version] - Urban, D.F.; Ambacher, O.; Elsässer, C. First-principles calculation of electroacoustic properties of wurtzite (Al,Sc)N. Phys. Rev. B
**2021**, 103, 115204. [Google Scholar] [CrossRef] - Ambacher, O.; Christian, B.; Feil, N.; Urban, D.F.; Elsässer, C.; Prescher, M.; Kirste, L. Wurtzite ScAlN, InAlN, and GaAlN crystals, a comparison of structural, elastic, dielectric, and piezoelectric properties. J. Appl. Phys.
**2021**, 130, 45102. [Google Scholar] [CrossRef] - Caro, M.A.; Zhang, S.; Riekkinen, T.; Ylilammi, M.; Moram, M.A.; Lopez-Acevedo, O.; Molarius, J.; Laurila, T. Piezoelectric coefficients and spontaneous polarization of ScAlN. J. Physics Condens. Matter
**2015**, 27, 245901. [Google Scholar] [CrossRef] [Green Version] - Tsubouchi, K.; Sugai, K.; Mikoshiba, N. AlN material constants evaluation and SAW properties on AlN/Al2O3 and AlN/Si. In Proceedings of the 1981 Ultrasonics Symposium, Chicago, IL, USA, 14–16 October 1981; pp. 375–380. [Google Scholar]
- Sotnikov, A.V.; Schmidt, H.; Weihnacht, M.; Smirnova, E.P.; Chemekova, T.Y.; Makarov, Y.N. Elastic and piezoelectric properties of AlN and LiAlO2 single crystals. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2010**, 57, 808–811. [Google Scholar] [CrossRef] - Kim, T.; Kim, J.; Dalmau, R.; Schlesser, R.; Preble, E.; Jiang, X. High-temperature electromechanical characterization of AlN single crystals. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2015**, 62, 1880–1887. [Google Scholar] [CrossRef] - Kazan, M.; Moussaed, E.; Nader, R.; Masri, P. Elastic constants of aluminum nitride. Phys. Status Solidi
**2007**, 4, 204–207. [Google Scholar] [CrossRef] - Watt, J.P.; Peselnick, L. Clarification of the Hashin-Shtrikman bounds on the effective elastic moduli of polycrystals with hexagonal, trigonal, and tetragonal symmetries. J. Appl. Phys.
**1980**, 51, 1525–1531. [Google Scholar] [CrossRef] - Mayrhofer, P.; Euchner, H.; Bittner, A.; Schmid, U. Circular test structure for the determination of piezoelectric constants of Sc
_{x}Al_{1−x}N thin films applying Laser Doppler Vibrometry and FEM simulations. Sens. Actuators A Phys.**2014**, 222, 301–308. [Google Scholar] [CrossRef] [Green Version] - Manna, S.; Talley, K.R.; Gorai, P.; Mangum, J.; Zakutayev, A.; Brennecka, G.L.; Stevanović, V.; Ciobanu, C.V. Enhanced Piezoelectric Response of AlN via CrN Alloying. Phys. Rev. Appl.
**2018**, 9, 034026. [Google Scholar] [CrossRef] [Green Version] - Nye, J.F. Physical Properties of Crystals: Their Representation by Tensors and Matrices; Oxford University Press: Oxford, UK, 1985. [Google Scholar]
- Qin, L.; Chen, Q.; Cheng, H.; Wang, Q.-M. Analytical study of dual-mode thin film bulk acoustic resonators (FBARs) based on ZnO and AlN films with tilted c-axis orientation. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2010**, 57, 1840–1853. [Google Scholar] [CrossRef] - Newnham, R.E. Properties of Materials: Anisotropy, Symmetry, Structure; Oxford University Press: Oxford, UK, 2005. [Google Scholar]
- Kong, L.; Zhang, J.; Wang, H.; Ma, S.; Li, F.; Wang, Q.-M.; Qin, L. Simulation study of MEMS piezoelectric vibration energy harvester based on c-axis tilted AlN thin film for performance improvement. AIP Adv.
**2016**, 6, 125128. [Google Scholar] [CrossRef] - Qin, L.; Wang, Q.-M. Analysis of dual-mode thin film bulk acoustic resonators based on polar c-axis tilted wurtzite gallium nitride. J. Appl. Phys.
**2010**, 107, 114102. [Google Scholar] [CrossRef] - Xie, M.-Y.; Tasnádi, F.; Abrikosov, I.A.; Hultman, L.; Darakchieva, V. Elastic constants, composition, and piezolectric polarization in InxAl1−xN: From ab initio calculations to experimental implications for the applicability of Vegard’s rule. Phys. Rev. B
**2012**, 86, 155310. [Google Scholar] [CrossRef] [Green Version] - Benchabane, S.; Gaiffe, O.; Salut, R.; Ulliac, G.; Laude, V.; Kokkonen, K. Guidance of surface waves in a micron-scale phononic crystal line-defect waveguide. Appl. Phys. Lett.
**2015**, 106, 81903. [Google Scholar] [CrossRef] [Green Version] - Wu, T.-C.; Wu, T.-T.; Hsu, J.-C. Waveguiding and frequency selection of Lamb waves in a plate with a periodic stubbed surface. Phys. Rev. B
**2009**, 79, 104306. [Google Scholar] [CrossRef] - Pennec, Y.; Laude, V.; Papanikolaou, N.; Djafari-Rouhani, B.; Oudich, M.; El Jallal, S.; Beugnot, J.-C.; Escalante, J.M.; Martínez, A. Modeling light-sound interaction in nanoscale cavities and waveguides. Nanophotonics
**2014**, 3, 413–440. [Google Scholar] [CrossRef] [Green Version] - Addouche, M.; Al-Lethawe, M.A.; Choujaa, A.; Khelif, A. Superlensing effect for surface acoustic waves in a pillar-based phononic crystal with negative refractive index. Appl. Phys. Lett.
**2014**, 105, 023501. [Google Scholar] [CrossRef] - Oudich, M.; Li, Y.; Assouar, B.; Hou, Z. A sonic band gap based on the locally resonant phononic plates with stubs. New J. Phys.
**2010**, 12. [Google Scholar] [CrossRef] - Achaoui, Y.; Khelif, A.; Benchabane, S.; Robert, L.; Laude, V. Experimental observation of locally-resonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars. Phys. Rev. B
**2011**, 83. [Google Scholar] [CrossRef] [Green Version] - Tian, Y.; Li, H.; Chen, W.; Lu, Z.; Luo, W.; Mu, X.; Wang, L. A Novel Love Wave Mode Sensor Waveguide Layer with Microphononic Crystals. Appl. Sci.
**2021**, 11, 8123. [Google Scholar] [CrossRef] - Serhane, R.; Belkhelfa, N.; Hadj-Larbi, F.; Merah, S.; Bakha, Y. Electrical Performances of a Surface Acoustic Wave Device With Inter Digital Transducers Electrodes in Local Resonances. J. Vib. Acoust. AMSE
**2021**, 143, 011009 [CrossRef]. [Google Scholar] [CrossRef] - Shao, L.; Maity, S.; Zheng, L.; Wu, L.; Shams-Ansari, A.; Sohn, Y.-I.; Puma, E.; Gadalla, M.; Zhang, M.; Wang, C.; et al. Phononic Band Structure Engineering for High- Q Gigahertz Surface Acoustic Wave Resonators on Lithium Niobate. Phys. Rev. Appl.
**2019**, 12, 14022. [Google Scholar] [CrossRef] [Green Version] - Laidoudi, F.; Boubenider, F.; Caliendo, C.; Hamidullah, M. Numerical Investigation of Rayleigh, Sezawa and Love Modes in C-Axis Tilted ZNO/SI for Gas and Liquid Multimode Sensor. J. Mech.
**2019**, 36, 7–18. [Google Scholar] [CrossRef] - Caliendo, C.; Hamidullah, M.; Mattioli, F. Finite Element Modeling and Synthesis of c-axis Tilted AlN TFBAR for Liquid Sensing Applications. Procedia Eng.
**2016**, 168, 1032–1035. [Google Scholar] [CrossRef]

**Figure 2.**Comparison between (

**a**) our calculated elastic constants C

_{ij}(GPa), and (

**b**) piezoelectric coefficients eij (C/m

^{2}), with data available from the literature.

**Figure 3.**Calculated dielectric constants ε

_{11}and ε

_{33}as a function of Sc concentration for w-Sc

_{x}Al

_{1-x}N.

**Figure 4.**(

**a**) Elastic constants, (

**b**) piezoelectric coefficient, and (

**c**) dielectric constants in (10

^{−11}F/m) of tilted w-Sc

_{0.375}Al

_{0.625}N.

**Figure 5.**(

**a**) PnC unit cell used for the dispersion curve calculation. (

**b**) First irreducible Brillouin zone.

**Figure 6.**Acoustic band structure of PnC AlN pillars with r = 3 µm, and h = 2.4 µm. The blue hatched area corresponds to the position of the acoustic band gap.

**Figure 7.**PnC band structures of lattice parameter a = 8 µm for AlN pillars of radius r = 3 µm and different values of h: (

**a**) h = 4 µm, (

**b**) h = 6 µm and (

**c**) h = 8 µm.

**Figure 8.**(

**a**) Evolution of the acoustic bandgaps (gaps map) as a function of the aspect ratio (h/a). (

**b**) Evolution of the center frequency f

_{c}and the bandgap width B

_{w}of the first and second bandgaps as a function of the height of the pillars of radius r = 2.6 µm.

**Figure 9.**Evolution of (

**a**) the bandgaps map, and (

**b**) the center frequency f

_{c}and bandgap width B

_{w}of the first and second bandgaps with Sc concentration (x).

**Figure 10.**Evolution of (

**a**) the bandgaps map, and (

**b**) the center frequencies f

_{c}and bandgap widths B

_{w}of the first and second bandgaps with the angle θ of tilted w-Sc

_{0.375}Al

_{0.625}N based PnCs structure.

**Figure 11.**(

**a**) Acoustic band structures of PnC w-Sc

_{0.375}Al

_{0.625}N pillars on half infinite substrate. (

**b**) Transmission through a finite PnC constituted of an 11-unit cell, with h = 6 µm, r = 2.6 µm, and a = 8 µm (red solid line) compared to the SAW through a non-structured surface (dashed lines). (

**c**) Map of the mechanical displacement fields of modes (A, B, C, D, and E).

**Figure 12.**(

**a**) 2D model used for the calculation of the transmission using the IDT. The PnC was set between the input and output IDTs. (

**b**) Insertion loss (S

_{21}) for different values of Sc concentration x of the non-structured surface.

**Figure 13.**Insertion loss (S

_{21}) with (red solid lines) and without (black solid lines) PnCs for different SAW device wavelengths, (

**a**) λ = 64 µm, (

**b**) λ = 30 µm, and (

**c**) λ = 20 µm closed to the eigenmodes A, C, and D.

**Figure 14.**(

**a**) Insertion loss (S

_{21}) of the SAW delay line without PnCs of w-Sc

_{0.375}Al

_{0.625}N. (

**b**) Insertion loss (S

_{21}) and frequency of SAW delay line without PnCs versus the c-axis tilted angle (θ°) of w-Sc

_{0.375}Al

_{0.625}N.

**Table 1.**Comparison of equilibrium lattice parameters (a, c) (Å) and elastic constants Cij (GPa) for w-Sc

_{x}Al

_{1-x}N compound (x = 0%, 12.5%, 25% and 37.5%) between our results (*) and the literature.

Material Constants | w-AlN | w-Sc_{x}Al_{1-x}N | ||||
---|---|---|---|---|---|---|

Th. | Exp. | x | 12.5% | 25% | 37.5% | |

a (Å) | 3.128 * | 3.11^{63} | 3.1842* | 3.2426 * | 3.3136 * | |

3.131^{64} | ||||||

c (Å) | 5.015 * | 4.98^{63} | 5.0489* | 5.0696 * | 5.065* | |

5.018^{64} | ||||||

C_{11} (GPa) | 376 * | 402.5 ± 0.5^{69} | 332.91 * | 302.56 | 282.64 * | |

432^{64} 374^{65} | 412.6 ± 0.05^{70} | 336.37^{65} | 305.68^{65} | 282.036^{65} | ||

378.8^{67} 345^{68} | 394 ^{71} | |||||

C_{12} (GPa) | 123 * | 135.6 ± 0.5^{69} | 126.86 * | 130.82 * | 121.45 * | |

170^{64} 129^{65} | 126.6 ± 0.5^{70} | |||||

128.9^{67} 125^{68} | 134 ^{71} | 121.95^{65} | 116.03^{65} | 110.44^{65} | ||

C_{13} (GPa) | 91* | 101 ± 2^{69} | 104.84 | 104.86 * | 112.06 * | |

147^{64} 101^{65} | 118.8 ± 0.9^{70} | 91.11^{65} | 83.43^{65} | 77.25^{65} | ||

96.1^{67} 120^{68} | 95^{71} | |||||

C_{33} (GPa) | 354* | 387.6 ± 1^{69} | 293.99 * | 251.91* | 220.94 * | |

390^{64} 351^{65} | 386.1 ± 4.5^{70} | 302.11^{65} | 255.33^{65} | 211.37^{65} | ||

357.5^{67} 395^{68} | 402^{71} | |||||

C_{44} (GPa) | 116* | 122.9 ± 0.5^{69} | 103.47 * | 96.09* | 87.29 * | |

155^{64} 112^{65} | 127.4 ± 0.9^{70} | 102.15^{65} | 97.489^{65} | 97.59^{65} | ||

112^{67} 118^{68} | 121^{71} |

**Table 2.**Comparison of calculated piezoelectric (e

_{15}, e

_{31}, e

_{33}) and dielectric constants (

**ε**

**,**

_{11}**ε**

**) for w-Sc**

_{33}_{x}Al

_{1-x}N compound between our results (*) and the literature.

Material Coefficient | w-AlN | w-Sc_{x}Al_{1-x}N | ||||
---|---|---|---|---|---|---|

Th. | Exp. | x | 12.5% | 25% | 37.5% | |

e_{31} (C/m^{2}) | −0.58 * | −0.54 ± 0.05^{9} | −0.624 * | −0.660 * | −0.686 * | |

−0.55^{64}–0.593^{65} | −0.6 ± 0.2^{69} | −0.625^{65} | −0.675^{65} | −0.743^{65} | ||

–0.424^{67}–0.58^{68} | −0.47 ± 0.2^{70} | |||||

e_{33} (C/m^{2}) | 1.45 * | 1.52 ± 0.43^{9} | 1.705 * | 2.026 * | 2.421 * | |

1.39^{64} 1.471^{65} | 1.34 ± 0.1^{69} | 1.70^{65} | 2.142^{65} | 2.788^{65} | ||

1.449^{67} 1.55^{68} | 2.09 ± 0.4^{70} | |||||

e_{15} (C/m^{2}) | −0.29 * | −0.30 ± 0.22^{9} | −0.311* | −0.306 * | −0.282 * | |

−0.30^{64}–0.313^{65} | −0.32 ± 0.05^{69} | −0.293^{65} | −0.256^{65} | −0.204^{65} | ||

–0.367^{67}−0.48^{68} | −0.24 ± 0.05^{70} | |||||

ε_{11} (10^{−11} F/m) | 8.3 * 8^{68} | 9.8 ± 07 ^{9} | 9.03 * | 9.75 * | 10.47 * | |

9 ± 0.01^{69} | ||||||

8.44 ± 0.1^{70} | ||||||

ε_{33} (10^{−11} F/m) | 9.75 * 9.5^{68} | 9.1 ± 0.3 ^{9} | 10.72 * | 11.96 * | 13.50 * | |

9.5 ± 0.01^{69} | ||||||

10.51 ± 0.1^{70} |

Material Constants | w-AlN | Al_{2}O_{3} |
---|---|---|

C_{11} (GPa) | 376 | 452 |

C_{12} (GPa) | 123 | 150 |

C_{13} (GPa) | 91 | 107 |

C_{33} (GPa) | 354 | 454 |

C_{44} (GPa) | 116 | 132 |

e_{15} (C/m^{2}) | 0.29 | |

e_{31} (C/m^{2}) | −0.58 | |

e_{33} (C/m^{2}) | 1.45 | |

ε_{11/}ε_{0} (C/m^{2}) | 8.31 | 11.07 |

ε_{33/}ε_{0} | 9.75 | 9.48 |

Young Modulus 10^{9} [Pa] | 364.05 | |

Poisson’s ratio | 0.24 | |

ρ (kg/m^{3}) | 3214.21 | 3870 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Arab, F.; Kanouni, F.; Serhane, R.; Pennec, Y.; Özer, Z.; Bouamama, K.
Electro-Acoustic Properties of Scandium-Doped Aluminum Nitride (*Sc _{x}Al_{1-x}N*) Material and its Application to Phononic Crystal-Coupled SAW Devices.

*Crystals*

**2022**,

*12*, 1431. https://doi.org/10.3390/cryst12101431

**AMA Style**

Arab F, Kanouni F, Serhane R, Pennec Y, Özer Z, Bouamama K.
Electro-Acoustic Properties of Scandium-Doped Aluminum Nitride (*Sc _{x}Al_{1-x}N*) Material and its Application to Phononic Crystal-Coupled SAW Devices.

*Crystals*. 2022; 12(10):1431. https://doi.org/10.3390/cryst12101431

**Chicago/Turabian Style**

Arab, Fahima, Fares Kanouni, Rafik Serhane, Yan Pennec, Zafer Özer, and Khaled Bouamama.
2022. "Electro-Acoustic Properties of Scandium-Doped Aluminum Nitride (*Sc _{x}Al_{1-x}N*) Material and its Application to Phononic Crystal-Coupled SAW Devices"

*Crystals*12, no. 10: 1431. https://doi.org/10.3390/cryst12101431