#### 2.1. The SAW Propagation in GaPO_{4}

The SAW propagating along the

x direction of a semi-infinite

y-rotated GaPO

_{4} substrate is a generalized SAW (GSAW) as it has three particle displacement components (the longitudinal

U_{1}, the shear horizontal and vertical,

U_{2} and

U_{3}) that rapidly decrease in depth. The GSAW phase velocity is strongly affected by the cut angle, as shown in

Figure 1 where the angular dispersion velocity curves of the SAW and BAW (the longitudinal BAW, LBAW, and the two transverse BAW, the fast shear, FSBAW, and the slow shear, SSBAW, having velocity equal to 4342.35, 3289.78 and 2451.96 m/s, respectively) are reported.

**Figure 1.**
The bulk acoustic wave (BAW) and generalized surface acoustic wave (GSAW) angular dispersion curves, and the GSAW K^{2} angular dispersion curves along (0° θ 0°) GaPO_{4} substrate.

**Figure 1.**
The bulk acoustic wave (BAW) and generalized surface acoustic wave (GSAW) angular dispersion curves, and the GSAW K^{2} angular dispersion curves along (0° θ 0°) GaPO_{4} substrate.

In

Figure 1, the angle θ means the amount of rotation around the crystal

x axes of a

y-cut GaPO

_{4} plate and is expressed in the Euler notation. The GSAW phase velocity along (0° θ 0°) GaPO

_{4} is lower than that of the SSBAW except for the 162° ≤ θ ≤ 169° range. While varying the cut angle and the velocity moving toward the SSBAW velocity, the shear horizontal amplitude U

_{2} of the GSAW decreases more and more slowly in the depth and the GSAW loses energy into the bulk. The electromechanical coupling coefficient,

K^{2}, is a measure of the electrical to acoustic energy conversion efficiency and it was calculated as 2 × (

v_{f} −

v_{m})/

v_{f}, being

v_{f} and

v_{m} the GSAW velocity along the free surface and the surface covered by a infinitesimally thin conductive film. The

K^{2} angular dispersion curve is also shown in

Figure 1. The highest

K^{2} value is equal to 1.22% and it is reached at θ = 110°; while the velocity moving toward the SBAW velocity, the GSAW

K^{2} approaches the zero value, in accordance with the increased

U_{2} contribution in the depth. The acoustic waves velocity calculations along GaPO

_{4} were performed by using Matlab and the software from McGill University [

8] in the lossless approximation; the GaPO

_{4} material data (the mass density, the elastic, piezoelectric, dielectric constants) as well as the thermal expansion coefficients and the third order temperature coefficients of the elastic constants, valid in the −50 °C <

T < 700 °C temperature range, are referred to [

7,

9,

10], and are those provided by Piezocryst Advanced Sensorics GmbH, which is an European GaPO

_{4} wafers suppliers.

#### 2.2. The Flexural Plate Waves (FPWs) Propagation in GaPO_{4}

The FPWs propagating along a thin plate consist in two sets of infinite modes: the symmetric waves, whose particle displacements are symmetric about the central plane of the plate, and the antisymmetric waves, whose displacements have odd symmetry about the central plane of the plate. For sufficiently thin plate, only the first symmetric and antisymmetric modes, S

_{0} and A

_{0}, propagate:

Figure 2 shows the dispersion curves of a few modes propagating along (0° 90° 0°) GaPO

_{4} plate as an example.

**Figure 2.**
Dispersion curves of the fundamental and higher order modes propagating along (0° 90° 0°) GaPO_{4} plate.

**Figure 2.**
Dispersion curves of the fundamental and higher order modes propagating along (0° 90° 0°) GaPO_{4} plate.

The A

_{0} mode is clearly identified by its reducing velocity as the plate thickness approaches zero. On the contrary, the dispersion curve of the S

_{0} mode is characterized by a low dispersion region, where itsvelocity is almost constant and

v ≈

v_{LBAW}, followed by a drop in velocity. The magnitude of the flat zone of the dispersion curve depends on the rotation angle θ, as can be seen in

Figure 3 where the S

_{0} velocity dispersion curves are shown for different rotating angles and for plate thickness normalized to the acoustic wavelength,

h/λ, up to 0.5. The low dispersion region reaches its largest amplitude for θ = 40°.

**Figure 3.**
S_{0} velocity dispersion curves for different rotating angles.

**Figure 3.**
S_{0} velocity dispersion curves for different rotating angles.

For very small thicknesses, h/λ << 0.1, the S_{0} velocity increases slowly and reaches a plateau where the velocity is close to the LBAW velocity. In this flat region, for plate thicknesses up to a value that depends on θ, hereafter mentioned h/λ_{AM}, the S_{0} mode transforms for first to the AM counterpart with constant and dominant longitudinal displacement. Subsequently, the mode transforms to the QL counterpart for increasing the plate thicknesses up to h/λ_{QLM}: The longitudinal displacement component is no longer constant through the thickness of the plate and is dominant over the other two components. For higher plate thicknesses, the S_{0} mode is characterized by three particle displacement components and, when the velocity approaches the shear horizontal BAW velocity, the U_{2} component largely dominates over U_{1} and U_{3}. For large h/λ > 1, the mode begins to act as a Rayleigh wave propagating on the surface of the plate: The mode is confined to the surface of the plate to a depth of a few wavelengths and the plate acts as a semi-infinite half space.

In the present paper, the S

_{0} and A

_{0} modes propagation along GaPO

_{4} plates is studied for two different coupling structures as depicted in

Figure 4. The configurations called Substrate/Transducer (ST) and Metal/Substrate/Transducer (MST) refer to a plate with one surface that sustain the excitation and detection of the acoustic waves through the interdigital transducers (IDTs), while the opposite surface is free or covered by a ground metal electrode. The FPWs

K^{2} depends very strongly on θ, on the electrical boundary conditions and on the plate thickness. With increasing the plate thickness, for

h/λ >> 1, the

K^{2} moves toward the

K^{2} of the GSAW that propagates along the

x direction of the

y-rotated cut GaPO

_{4} semi-infinite substrate.

**Figure 4.**
The Substrate/Transducer (ST) and Metal/Substrate/Transducer (MST) coupling configurations.

**Figure 4.**
The Substrate/Transducer (ST) and Metal/Substrate/Transducer (MST) coupling configurations.

The

K^{2} dispersion curves of the S

_{0} and A

_{0} modes were studied for the ST and MST structures by calculating the perturbation of the phase velocity when the tangential electric field component is shorted out at the plate surface. For each configuration

K^{2} was approximated to be 2 × [(

v_{f} −

v_{m})/

v_{f}], where

v_{m} and

v_{f} are the phase velocities along the structures with and without the IDTs, under the assumption that a perfect conductor is placed at both the IDT and the ground electrode position. The

K^{2} dispersion curves of each configuration were calculated for different plate thicknesses normalized to the acoustic wavelength,

h/λ, in the range 0.0001 to 2, with 0.0001

h/λ step, in the approximation of infinitesimally-thin perfectly-conductive grounded electrode and IDTs.

K^{2} values much higher than that of the SAW travelling in the same direction and crystal plane can be reached in thin plates (

h/λ << 1), for both S

_{0} and A

_{0} modes, even for that cut angles for which the SAW has a

K^{2} = 0%. The

K^{2} angular dispersion curves of the S

_{0} mode in both the ST and MST configurations have similar behavior: The

K^{2} reaches a peak for small plate thickness (at about

h/λ ~ 0.1 and 0.4, respectively), then it resets and increases again, and finally it asymptotically reaches the

K^{2} of the SAW.

Figure 5 shows, as an example, the S

_{0} mode

K^{2} dispersion curves of the two configurations in (0° 85° 0°)

x-GaPO

_{4}. The

K^{2} dispersion curve of the MST configuration has a second broad peak at

h/λ ~ 2 that is higher than the

K^{2} of the SAW: With increasing the plate thickness, the

K^{2} decreases and then, for

h/λ >> 1, it asymptotically reaches the

K^{2} of the SAW travelling on the free surface of the plate.

**Figure 5.**
The S_{0} mode K^{2} dispersion of the two configurations in (0° 85° 0°) x-GaPO_{4}.

**Figure 5.**
The S_{0} mode K^{2} dispersion of the two configurations in (0° 85° 0°) x-GaPO_{4}.

Figure 6 shows the

K^{2} angular dispersion curve of the SAW travelling along the free plate surface, and the angular dispersion curves of the maximum

K^{2} value achievable by the S

_{0} mode for the two coupling configurations. The

K^{2} peak value of the ST configuration is higher than the

K^{2} of the SAW for θ ≤ 90° and >150°, while that of the MST configuration is higher only for θ ranging from 0° to 30° and θ >140°.

**Figure 6.**
The angular dispersion curves of the maximum K^{2} value achievable with the two S_{0} mode coupling configurations and the SAW K^{2} angular dispersion.

**Figure 6.**
The angular dispersion curves of the maximum K^{2} value achievable with the two S_{0} mode coupling configurations and the SAW K^{2} angular dispersion.

The highest coupling efficiency of the S_{0} mode is around 1.2%, and it is reached by the ST configuration at h/λ ~ 0.11.

As an example,

Figure 7 shows the field profile of the shear mode propagating at

v = 3300 m/s along the (0° 75° 0°)

x-GaPO

_{4} plate with

h/λ = 0.11, for which the conditions U

_{2} >> U

_{1}, U

_{3} across the depth are satisfied, and

v ~

v_{SHBAW}. This type of mode is largely applied for sensing applications in liquid environments [

11,

12,

13] thanks to its right polarization, but it is disadvantageous compared to the AMs or QLMs as it travels at a lower velocity.

**Figure 7.**
The field profile of the shear horizontal (SH) wave travelling along (0° 75° 0°) x-GaPO4 plate with h/λ = 0.7.

**Figure 7.**
The field profile of the shear horizontal (SH) wave travelling along (0° 75° 0°) x-GaPO4 plate with h/λ = 0.7.

The A

_{0} mode is far more efficient than the S

_{0} and its most efficient of the two configurations is the MST for all the θ values from 0° to 180°. The

K^{2} dispersion curves of the A

_{0} mode depends on θ, on the electrical boundary conditions and on the plate thickness: These curves have a peak value at

h/λ << 1, and then asymptotically reach the

K^{2} of the SAW with increasing the plate thickness. As an example,

Figure 8 shows the

K^{2} dispersion curves of the two A

_{0}-based configurations on (0° 95° 0°)

x-GaPO

_{4}.

**Figure 8.**
The K^{2} of the two A_{0} coupling configurations on (0° 95° 0°) x-GaPO_{4}.

**Figure 8.**
The K^{2} of the two A_{0} coupling configurations on (0° 95° 0°) x-GaPO_{4}.

Figure 9 shows the angular dispersion of the

K^{2} peaks of the A

_{0} mode for the two coupling configurations.

K^{2} as high as 1.9% and 2.64% can be reached by the ST and MST configurations for θ = 95°, and for plate thickness equal to

h/λ = 0.458 and 0.31.

**Figure 9.**
The angular dispersion of the K^{2} peaks of the A_{0} mode for the two coupling configurations.

**Figure 9.**
The angular dispersion of the K^{2} peaks of the A_{0} mode for the two coupling configurations.

The A_{0} mode, while being elliptically polarized, can travel along thin membranes that are in contact with a liquid as a consequence of its dispersive phase velocity that can be designed to be lower than the liquid medium compressional velocity, at the proper plate thickness. The phase velocity corresponding to the K^{2} peak values is 1888 and 1583 m/s for the ST and MST configurations, respectively. Thus, a little decrease in K^{2} has to be accepted to reduce the velocity of the A_{0} mode up to a value lower than the velocity of the fluid environment (about 1500 m/s for water). Another benefit of the low-velocity A_{0} mode is that it allows for inexpensive signal processing equipment to be used, due to its low resonant frequency f = v/λ.

#### 2.3. The Anisimkin Jr. Modes

In references [

14,

15] the peculiarities of the AMs are so defined: they have a velocity very close to that of the longitudinal BAW and have a dominant longitudinal displacement component

U_{1} with constant amplitude along the whole depth of the plate, being the shear components,

U_{2} and

U_{3}, at least 10 times less than

U_{1} at any plate depth. The propagation of the AMs was investigated by studying the field profile of the modes that propagate along the plate with a phase velocity close to that of the LBAW: Plate thicknesses up to

h/λ = 2 were studied. As an example,

Figure 10 shows the field profile (the three displacement components normalized to the U

_{1} surface value

vs. the plate depth) of the AM propagating along the (0° 60° 0°)

x-GaPO

_{4} plate with

h/λ = 0.11.

**Figure 10.**
The field profile of the Anisimkin Jr. mode (AM) propagating at velocity v = 4191 m/s along a (0° 60° 0°) x-GaPO_{4} plate h/λ = 0.11 thick.

**Figure 10.**
The field profile of the Anisimkin Jr. mode (AM) propagating at velocity v = 4191 m/s along a (0° 60° 0°) x-GaPO_{4} plate h/λ = 0.11 thick.

Figure 11 shows the normalized plate thicknesses for which both the conditions

U_{1} = 1,

U_{2},

U_{3} ≤ 0.1 across the plate depth and

v ~

v_{LBAW} are satisfied and the maximum

K^{2} value achievable in this thickness range, for the ST and MST configurations, as a function of the substrate

y-rotation angle. No other plate thickness range suitable for the AMs propagation was found in the 0 to 2 range of

h/λ.

As can be seen, the ST configuration is the most favorable as it ensures the highest K^{2} at the largest plate thickness: the latter characteristic is particularly important as it is difficult to fabricate and handle a device implemented on a thin and fragile plate.

**Figure 11.**
The AMs plate thickness range, 0–h/λ_{AM}, and the peak K^{2} of the two configurations vs. the rotation angle.

**Figure 11.**
The AMs plate thickness range, 0–h/λ_{AM}, and the peak K^{2} of the two configurations vs. the rotation angle.

The temperature coefficient of velocity (

TCV),

TCV = (1/

T °C)(

v_{T} _{°C} −

v_{20} _{°C})/v

_{20} _{°C}, that is linked to the

TCD by the relation

TCD = α −

TCV, being α the thermal expansion coefficient of the plate, was calculated over the plate thickness range, 0.001–

h/λ

_{AMs}, over which the AMs propagate. The calculated

TCV is referred to 20 °C, being

v_{T} _{°C} and

v_{20} _{°C} the wave velocity at a certain temperature value between −20 and 420 °C, and at 20°C, respectively. The relative velocity shift (

v_{T} _{°C} −

v_{20} _{°C})/

v_{20} _{°C} vs. temperature curves, calculated for different

h/λ values in the 0.001–

h/λ

_{AMs} range for each

y-rotation angle, were then fitted by a quadratic function.

Figure 12 shows the first and second order coefficients A

_{1} and A

_{2} of the quadratic fit: The bar centered on each point refers to the coefficient evaluated at the upper (

h/λ

_{AMs}) and lower (0.001) plate thickness range. As can be seen in

Figure 12, for θ ≥ 55°, the two coefficients are poorly affected by the plate thickness change over the 0.001–

h/λ

_{AMs}.

**Figure 12.**
The first and second order coefficients, A_{1} and A_{2}, of the temperature coefficient of velocity (TCV) quadratic fit.

**Figure 12.**
The first and second order coefficients, A_{1} and A_{2}, of the temperature coefficient of velocity (TCV) quadratic fit.

In the 60° to 90° range, the two coefficients A_{1} and A_{2} reach very small values, hence zero or very low TCD structures based on the AMs propagation can be fabricated.

The PFA is related to the incremental slope of the inverse velocity surface. The PFA of the AMs propagating along the x-axis of the unmetallized and metallized surface of the GaPO_{4} plates was theoretically investigated in order to evaluate the effect of an electrical shorting layer on the plate surface on the PFA. In both the two cases the PFA is very small, in the 0 to 0.09 range, and it is poorly affected by the presence of a shorting layer on one plate surface.

#### 2.4. The Quasi Longitudinal Modes

In the present paper, a mode is attributed to the QL-family if

U_{1} is not constant through the plate, and

U_{2} and

U_{3} are less than 20% of

U_{1} at any depth. As an example,

Figure 13 shows the field profile of the QLM propagating along the (0° 0° 0°)

x-GaPO

_{4} plate at

v = 4097 m/s, being the plate thickness equal to

h/λ = 0.22.

**Figure 13.**
The field profile of the quasi longitudinal mode (QLM) propagating along the (0° 0° 0°) x-GaPO_{4} plate with h/λ = 0.22 at v = 4097 m/s.

**Figure 13.**
The field profile of the quasi longitudinal mode (QLM) propagating along the (0° 0° 0°) x-GaPO_{4} plate with h/λ = 0.22 at v = 4097 m/s.

In

Figure 14, the first and second order coefficients of the quadratic fit of the

TCV vs. T °C of the QLMs are shown, being these coefficients calculated over the

h/λ

_{AMs}–

h/λ

_{QLM} range, for each y rotation angle.

As compared with the AMs, the coefficients A_{1} and A_{2} of the QLMs are a little bit more temperature sensitive than those of the AMs.

As for the AMs, the PFA of the QLMs propagating along the x axis of y rotated GaPO_{4} plates is very small, in the 0 to 0.09 deg range, and poorly affected by the presence of a metal shorting layer on the plate surface.

**Figure 14.**
The first and second order coefficients of the quadratic fit of the TCV of the QLMs propagating along the (0° θ 0°) GaPO_{4} plate.

**Figure 14.**
The first and second order coefficients of the quadratic fit of the TCV of the QLMs propagating along the (0° θ 0°) GaPO_{4} plate.