Data analysis

FB F. Bencivenga RM R. Mincigrucci FC F. Capotondi LF L. Foglia DN D. Naumenko AM A. A. Maznev EP E. Pedersoli AS A. Simoncig FC F. Caporaletti VC V. Chiloyan RC R. Cucini FD F. Dallari RD R. A. Duncan TF T. D. Frazer GG G. Gaio AG A. Gessini LG L. Giannessi SH S. Huberman HK H. Kapteyn JK J. Knobloch GK G. Kurdi NM N. Mahne MM M. Manfredda AM A. Martinelli MM M. Murnane EP E. Principi LR L. Raimondi SS S. Spampinati CS C. Spezzani MT M. Trovò MZ M. Zangrando GC G. Chen GM G. Monaco KN K. A. Nelson CM C. Masciovecchio

This protocol is extracted from research article:

Nanoscale transient gratings excited and probed by extreme ultraviolet femtosecond pulses

**
Sci Adv**,
Jul 26, 2019;
DOI:
10.1126/sciadv.aaw5805

Nanoscale transient gratings excited and probed by extreme ultraviolet femtosecond pulses

Procedure

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 (*I*_{sig}) 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 by$${I}_{\text{sig}}\sim {\mid {A}_{\text{th}}\text{exp}(\u2010\mathrm{\Delta}t/{\mathrm{\tau}}_{\text{th}})+{\mathrm{\Sigma}}_{\mathrm{i}}{A}_{\mathrm{i}}\text{exp}(\u2010\mathrm{\Delta}t/{\mathrm{\tau}}_{\mathrm{i}})\text{cos}(2{\mathrm{\pi}\mathrm{\nu}}_{\mathrm{i}}\mathrm{\Delta}t)\mid}^{2}$$(1)where *A*_{th} and τ_{th} are the amplitude and time decay of the thermal relaxation, respectively, while *A*_{i} and τ_{i} are those of the coherent phonon excitations. Therefore, the exponential decay of the signal is characterized by a time constant τ_{th}/2 (red lines in Fig. 2), while the decay rate of the ν_{i} modulations, τ_{th}^{−1} + τ_{i}^{−1}, is determined by both the thermal and phonon decay. Consequently, the oscillations at ν_{i} 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 τ_{th} (*38*). According to Eq. 1, the signal should also contain components oscillating at 2ν_{i}, with decay rates of 2/τ_{i}, 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 Δ*t* and signal-to-noise ratio for determining the 2ν_{i} oscillating term. We estimated τ_{th} from the best fit of the data to Eq. 1 with *A*_{i} = 0 (red lines in Figs. 2, A to C, and 4), which resulted, respectively, in τ_{th} = 750 ± 190 ps, 370 ± 50 ps, and 42 ± 9 ps for *L*_{TG} = 110, 85, and 28 nm (for Si_{3}N_{4}) and in τ_{th} = 73 ± 12 ps for Si at *L*_{TG} = 110 nm. The value of the thermal diffusivity, *D*_{th} = 470 ± 90 nm^{2}/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 Si_{3}N_{4} membranes show large variations, even in samples from the same batch (*24*, *35*, *39*, *40*). For instance, values as different as 500 nm^{2}/ns (*24*) and 2500 nm^{2}/ns (*40*) were reported for different fabrication procedures, compositions, residual stress, thickness, etc.

After subtraction of the corresponding *A*_{th} exp{−2Δ*t*/τ_{th}} 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 Δ*t* range are needed to investigate these features.

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