To quantify the parameters characterizing the signal waveform, we used a basic data analysis, having the advantage of not relying upon specific modeling and consisting in an exponential function to estimate the slow decay (τ) and a Fourier transform to extract the oscillation frequency (ν). The TG signal intensity (Isig) is quadratic with respect to the strain amplitude changes induced by the TG (4, 6, 18, 21, 22), and the thermoelastic response can be approximated byIsigAthexp(Δt/τth)+ΣiAiexp(Δt/τi)cos(2πνiΔt)2(1)where Ath and τth are the amplitude and time decay of the thermal relaxation, respectively, while Ai and τi are those of the coherent phonon excitations. Therefore, the exponential decay of the signal is characterized by a time constant τth/2 (red lines in Fig. 2), while the decay rate of the νi modulations, τth−1 + τi−1, is determined by both the thermal and phonon decay. Consequently, the oscillations at νi in the signal appear as damped even if the coherent phonon excitations do not decay at all in the probed time scale. This is the expected situation for both samples, because the time decay of LA phonons is expected to be about one order of magnitude longer than τth (38). According to Eq. 1, the signal should also contain components oscillating at 2νi, with decay rates of 2/τi, that would persist beyond the thermal relaxation. We investigated this behavior in our previous experiment with an optical probe (18); however, in the present study, our measurements lack a sufficient range in Δt and signal-to-noise ratio for determining the 2νi oscillating term. We estimated τth from the best fit of the data to Eq. 1 with Ai = 0 (red lines in Figs. 2, A to C, and 4), which resulted, respectively, in τth = 750 ± 190 ps, 370 ± 50 ps, and 42 ± 9 ps for LTG = 110, 85, and 28 nm (for Si3N4) and in τth = 73 ± 12 ps for Si at LTG = 110 nm. The value of the thermal diffusivity, Dth = 470 ± 90 nm2/ns, estimated from data in Fig. 3B, is in the range of the values reported in the literature (24). However, we have to notice that the thermal properties of Si3N4 membranes show large variations, even in samples from the same batch (24, 35, 39, 40). For instance, values as different as 500 nm2/ns (24) and 2500 nm2/ns (40) were reported for different fabrication procedures, compositions, residual stress, thickness, etc.

After subtraction of the corresponding Ath exp{−2Δtth} terms from the signal, the residual waveforms are Fourier-transformed to determine their spectral content (the results are shown in fig. S1F). The main peak matches the frequency expected for LA phonons, as discussed in the main text (see Fig. 3A), while weaker peaks at low frequencies and at 2ν can be perceived above the noise level. However, higher quality data and an extended Δt range are needed to investigate these features.

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