#### 4.1. Residual Temperature Sensitivity

The thermal coefficients of expansion (TCE) of LN and Si are different. They are respectively 7.5 ppm/K and 2.6 ppm/K, for LNYZ and Si, in the propagation direction. This generates thermally induced stress and strain fields in the structure, when the temperature changes.

Strain and stress fields affect the wave propagation properties of SAW devices. As mentioned already, that is (of course) why they can be used as strain and pressure sensors. That is also why growing thermo-mechanical strain fields result in residual temperature sensitivity, for the LN/Si sensors. This effect was accurately computed and accounted for, in previous work [

24]. However, the temperature sensitivity of our sensor, which should be zero by design, is also influenced by any imperfection in the mounting of the device in its housing and by residual, unpredictable stress and strain fields after wafer-bonding. Consequently, it appears almost impossible to solve this issue other than by directly measuring the temperature and correcting the pressure readings accordingly.

A first, straightforward solution consists in measuring the temperature using an additional temperature sensor. To accurately measure the temperature at the pressure sensor location, both sensors need to be as close as possible to each other. A second R-DL could be used, next to the first one, to measure temperature only. However, such a solution would be suboptimal as it would require additional space and, more important, would reduce the reading range of the system. Indeed, the energy provided by the reader would be shared between the two R-DLs, which are connected to the same antenna.

Another, more elegant solution, would consist in extracting the temperature information from the existing design. As mentioned and used above, the combination of the properties of Sections S1, S2 and S3 provides temperature-compensated pressure readings, using S = ϕ(S2) − [ϕ(S1) + ϕ(S3)]. On the other hand, the quantity U = ϕ(S2) + [ϕ(S1) + ϕ(S3)] would provide temperature readings, however with residual pressure sensitivity (the pressure sensitivity of S1 + S3 is not exactly opposite to the pressure sensitivity of S2, even in the ideal case). The expected accuracy of these temperature readings can be estimated as follows. The achieved accuracy in time domain strongly depends on the bandwidth of the system and on the Signal-to-Noise Ratio (SNR). For an 80 MHz bandwidth, an accuracy of 0.3–0.6 ns is achievable. In our case, the achievable temperature accuracy would therefore be close to ~1–2 °C. In the worst case, this would still translate into a residual pressure uncertainty of ~2 mbar (~1.5 Torr), which is too large a value for practical applications.

Nevertheless, the obtained approximate temperature value would still provide essential information that could be refined using phase analysis, and additional information from one extra reflector. The extra reflector (Rph) shall be located between R1 and R2, and far enough from the edge of the membrane to stay unaffected by pressure. The propagation time between R1 and Rph shall be set to ~250 ns. In this configuration, a temperature variation from 0 °C to 50 °C would make the value of the differential phase ϕ_{(Rph–R1)} change by 6π. This would make it possible to achieve high temperature accuracy within the temperature operation range, as the phase of a given peak can itself be measured with high accuracy (±0.05–0.1 rad). With a total differential phase shift of 6π, the achievable temperature accuracy would be close to ±0.12–0.25 °C. However, this would generate phase ambiguity issues, as one phase value would correspond to three possible temperature values. The approximate temperature information obtained before would finally be used here, to resolve the ambiguity. Such a solution should eventually make it possible to achieve high accuracy temperature readings (better than ±0.25 °C), which would, in any case, be sufficient to compensate for the effect of the residual temperature sensitivity on pressure readings. It will be the purpose of future work to implement this solution.

#### 4.2. Membrane Buckling

The thermally induced stress and strain fields can also result in a more problematic change in the device behavior, as they can lead to the buckling of the membrane above a given temperature threshold. We believe that buckling explains the discontinuity observed at ~43 °C, in the phase versus temperature curves of peaks R2, R3 and R4, obtained for the selected sensor (see

Figure 7c). The buckling hypothesis is supported by preliminary profilometry measurements of the shape of the membrane versus temperature, performed at zero pressure (i.e., the pressure stayed constant and equal to 1 bar, on both sides of the membrane, during the whole experiment) (see

Figure 8). The deformation clearly increases with temperature, and sudden shape changes can be observed, between 44 °C and 50 °C. The already deformed state, at ~44 °C is due to residual stress, which is generated in the LN/Si structure after wafer bonding, and concentrates in the membrane after the underlying Si is etched away. Interestingly, the shape becomes more regular at higher temperature, when the thermally induced deformation gets larger. Future work includes thorough characterization of the membrane shape at lower temperatures, in the range 20–44 °C.

Possible solutions to the buckling issue include thermal annealing (to reduce residual stress and shift the buckling point outside the temperature operating range of the sensor), and the pre-stressing of the membrane, to operate beyond the buckling point of the membrane, i.e., in a ‘buckling-free’ deformation range. A practical implementation of this solution would require the membrane to be equipped with a sealed vacuum cavity, deposited on top of the SAW track, and therefore located on the same side of the membrane as the IDT and reflectors. Vacuum would be preferred to any encapsulated reference pressure, as changes in temperature would also affect the reference pressure. The cavity should therefore be perfectly sealed, with no negative impact of the cavity walls on the propagating SAWs. Such a solution would certainly be costly, and difficult to implement.

It would therefore be better to solve the issue by changing or adjusting existing design parameters like, among others, the shape and thickness of the membrane. To do that, it is necessary to develop a simulation model first, which accounts for the observed buckling phenomenon. We built such a 3D model using Comsol Multiphysics. The model was used to perform thermo-mechanical studies of the membrane. The model takes advantage of the symmetries of the geometry, material and loads, which makes it possible to represent one fourth of the entire structure only. This simplification is acceptable, as the first buckling mode presents the same symmetry. Further simplification is achieved, by not taking the IDT, reflectors and Au/Cr bonding layer into account. In the model, the thickness of the membrane is exactly 30 μm. The boundary conditions include fixed displacements at the bottom of the substrate. The temperature is considered homogeneous, in the entire structure. No residual stress is considered, in the model. Therefore, the membrane is perfectly flat, at the initial temperature T0. An eigenvalue analysis made it possible to compute the buckling temperature as well as the deformed shape of the membrane, after buckling. The computations yielded a buckling temperature of 44.5 °C, which constitutes a bifurcation point. Above this point, the membrane can deform (buckle) in one of the two possible directions, up and down. As both directions are equivalent, a solution must be found to impose the deformation in one chosen direction, numerically. This was achieved by fixing a level of deformation for the membrane (i.e., by applying a pressure or a force on it), before letting the FEM software look for the temperature that would generate this pre-defined deformation, in the buckled state (see

Figure 9a,b). The results show a maximal displacement of 35 μm at 60 °C. This is 40% larger than the experimentally observed displacement (see

Figure 8). The stress and strain fields along the acoustic path of the sensor were also computed. Therefore, it became possible to compute the additional (unwanted) sensor sensitivity to temperature in the buckled state, following the method described in Reference [

24]. The resulting temperature sensitivity (after buckling) in phase domain is shown in

Figure 9c. It is also compared to the measured ‘phase vs. temperature’ characteristic curve. The model overestimates the temperature sensitivity after buckling, but it gives the right tendency. Therefore, it is already possible to use it for design improvement purposes. Future work will include the testing of different membrane shapes (including elliptical membranes), to shift the buckling point beyond 45 °C.