We present an evaluation of attenuation of elastic waves in the GHz range, based on the decay of thickness-stretch resonances of plates. We measured the elastic response of micro-scale aluminum plates, using a laser-ultrasound technique. The thermo-elastic laser excitation provides significant coupling into thickness-stretch modes in the frequency range above 1.5 GHz. To suppress interference from other resonant and transient modes, we used an excitation spot size well above the plate thickness and applied signal processing in the time and frequency domain. We adapted existing theory on the decay of thickness-shear modes to apply for thickness-stretch modes, validated the derived theory with simulations, and applied it to experiments. A set of samples with different thicknesses in the range of 1.1–2.15 μm allowed us to obtain damping values in the corresponding frequency range of 1.5–3 GHz.

Elastic waves propagating in solids decay over space and time. In addition to diffraction, thermo-elastic damping1,2 and microscopic mechanisms, like grain boundary scattering,3–7 play an important role for the attenuation of ultrasound. A common approach is to account for these mechanisms with effective attenuation coefficients. For macroscopically isotropic materials, two frequency dependent coefficients, αL(f) and αT(f), are required to describe the exponential decay of longitudinal and transverse waves, respectively.

Micro-acoustic filters used for mobile communication typically operate at low GHz frequencies, with new generations (like 5G) extending the range. A reliable, direct measurement of attenuation at these frequencies will help to accelerate the design of new devices and optimize material processes. Our investigation is focused on αL(f), the attenuation of longitudinal waves, which are predominant in bulk acoustic wave filters. We present our results as a quality factor Q(f)=f/2αL.

Laser-ultrasound (LUS) methods can operate in the GHz frequency regime8–10 and work contact free, and thus influences on the excited elastic waves are minimal. LUS was used to evaluate elastic wave attenuation in the MHz range, by comparing the spectral content of successive echoes of a longitudinal pulse in plane-parallel samples.11 Recently, the measurement of attenuation of surface acoustic waves (SAWs) was reported6 in a similar frequency range. For frequencies from 10 GHz upwards, picosecond ultrasonics has been applied to measure acoustic attenuation.12,13 However, there is a lack of direct measurement methods in the technologically relevant range from 110 GHz. In an early stage of this work, we found that we can resolve plate resonances around 2 GHz with a high signal-to-noise ratio (SNR) and frequency resolution in micrometer scale samples. The large resonant amplitudes in comparison to pulse measurements and the simple optical setup in contrast to SAW measurements, which require scanning and close to diffraction limited spot sizes for the frequency range of interest,14 motivated the presented approach. Local response-spectra of plates can be accessed with LUS10,15–21 and contain peaks associated with thickness resonances, showing at frequencies f when standing longitudinal or shear waves are present across the thickness h of the plate. For thickness-stretch modes, the condition is fLnh=ncL/2, where n=1,2,3, and cL is the longitudinal sound velocity.22 Prada et al. and Laurent et al. analyzed the temporal decay of the surface displacement amplitudes in the center of a finite sized laser source, which couples into plate resonances.23,24 They derived approximate temporal power laws for the decay of thickness-shear resonances and zero-group-velocity (ZGV) resonances for plates with negligible material attenuation and showed that material attenuation can be accounted for by an exponential decay factor eαZGVt. It was shown that the decay behavior is dominated by the power law in the first instances after excitation and then transitions into the exponential decay associated with material attenuation,23,25 which can also be extracted from measurements. In contrast to ZGV modes, which exhibit longitudinal and shear contributions, thickness-stretch modes have pure longitudinal polarization. Thus, their attenuation links directly to αL and its equivalent representation as a quality factor Q=fLn(2αL)1.

In the following, we report the predominant excitation of thickness-stretch resonances at GHz frequencies in micrometer scale plates and the detection of the response with LUS. We adapt an existing theory to approximate their decay behavior and validate the found relations with numerical simulations. We identify issues that persist when evaluating real world samples and provide strategies to avoid these in future experiments. Based on our state-of-art method, we provide Q-values for longitudinal waves in the investigated material from 1.5 to 3 GHz.

For the plate response measurements, we adapted a LUS setup consisting of a Michelson interferometer and an intensity modulated excitation laser to work in the GHz regime.26 The principle of the setup is described in detail elsewhere.27,28 Intensity modulated sources in combination with low-noise narrow-band detection6,9,29,30 achieve a high SNR at low excitation intensities, which is crucial when investigating delicate micro-scale samples. The used excitation power in all experiments was between 0.8 and 1 W, with a modulation depth (ratio of peak-to-peak to maximum) of 0.25. The vibrometer detects the out-of-plane displacement component uZ of the sample surface.

The geometry of the investigated samples—a micrometer scale aluminum plate sustained on two edges—and the confocal arrangement of the excitation laser and vibrometer are shown in Fig. 1(a). The details on the production of the samples are included in the supplementary material. We evaluated six samples with thicknesses ranging from 1.1 to 2.15 μm. The width of the samples is 100 μm, and the length varies between 200 and 800 μm. The surface of the samples was mirror-like, and we neglect any influence of surface roughness in this study. A simple microscope coupled to the LUS setup provided a top view of the samples and allowed us to position the detection laser spot [see the image in Fig. 1(a)].

FIG. 1.

(a) LUS measurement scheme and geometry of samples. (b) Measured response of a 1.08 μm aluminum plate, showing resonant peaks. (c) Calculated dispersion curves, providing an assignment of the peaks in (b) to mode cutoffs and a ZGV mode.

FIG. 1.

(a) LUS measurement scheme and geometry of samples. (b) Measured response of a 1.08 μm aluminum plate, showing resonant peaks. (c) Calculated dispersion curves, providing an assignment of the peaks in (b) to mode cutoffs and a ZGV mode.

Close modal

Figure 1(b) shows the measured response spectrum of a central point on a sample with thickness h = 1.08 μm. Here, the full-width-half-maximum (FWHM) diameters of both the Gaussian excitation and detection spots were below 3 μm. We obtain peaks associated with thickness-stretch resonances (L1 and L2) at f1=cL/(2h) and f2=cL/h. These correspond to cut-off frequencies in the dispersion curves (calculated for a 1.08 μm plate with longitudinal sound velocity cL=6385 m/s, transverse sound velocity cT=2985 m/s, and density ρ = 2780 kg/m3), shown in Fig. 1(c). The amplitudes of the L1 and L2 resonances achieved in this frequency range with thermo-elastic excitation are remarkably large, compared to measurements on thicker plates, where these modes are often not visible at all (see, for example, Ref. 24) However, the largest peak in the spectral response in Fig. 1(b) is the ZGV peak around 2.6 GHz, which corresponds to the local minimum in the dispersion curves.

While the ZGV peak is useful for many non-destructive testing applications,10,16–21 for the current purpose, we aim to selectively detect only the L1 thickness mode, with a minimum of contributions from other modes. We, therefore, suppress coupling into the ZGV mode and other modes by enlarging the excitation spot size significantly above the ZGV wavelength.25,31

We corroborate the effect of enhanced coupling into the desired L1 resonance at high frequencies and the effect of the spot size on the coupling into the interfering ZGV resonance with a numerical model and experiments. As a model, we use the coupled thermal and elastic differential equations for an infinitely extended plate, represented in cylindrical symmetry.32 The laser excitation is represented by a heat flux with Gaussian profile Aexp(r/R) in the thermal boundary conditions. The spot size is defined by the radius R at which the light intensity drops to 1/e. Material attenuation was accounted for by using complex valued sound velocities. An analytical solution for the out-of-plane displacement is found in the Hankel-domain and then transformed into the spatial domain by numerical integration.

The obtained spectra of out-of-plane surface displacement at the epicenter for a thick plate (h = 108 μm, blue line) and a thin plate according to our experiments (h = 1.08 μm, red line) with a (small) excitation spot size of R=2.2h for both cases are shown in Fig. 2(a). Due to the normalization of the frequency axis with h, the peaks coincide in this plot and can be compared. Note the large relative increase in the L1 peak at about 3.2 MHz mm for the thinner plate. As expected, the dominant ZGV peak in the spectrum disappears for the response shown in green, where the spot size was increased by a factor of 10. Also, contributions from non-resonant Lamb modes, which clearly show as a base-level in the response spectra for small spot sizes, are suppressed almost completely. An additional small peak appears around 2.95 MHz mm, which corresponds to a thickness-shear resonance and is buried within the ZGV response for the cases with small spot sizes. The large amplitude of the thickness-stretch mode in relation to the thickness-shear mode is in contrast to the reported behavior of thicker plates.24 We ascribe this to a significant out-of-plane component of the thermo-elastic source in the investigated frequency range. The measurements in Fig. 2(b) confirm the effective suppression of unwanted modes by increasing the spot size and a good SNR achievable for L1 resonances. The configuration producing the green curve in Fig. 2(b) used an excitation spot with R=1015 μm and was used in all further measurements.

FIG. 2.

(a) Simulated response spectra; blue (h = 108.0 μm, R=2.2h): almost no L1 peak; red (h = 1.08 μm, R=2.2h): dominant ZGV peak and significant L1 peak; green (h = 1.08 μm, R=22.1h): dominant L1 peak. (b) Measured response spectra, red (h = 1.08 μm, R2.2h): dominant ZVG peak and significant L1 peak green (h = 1.08 μm, R 10–15 h, and suppressed ZGV peak and dominant L1 peak.

FIG. 2.

(a) Simulated response spectra; blue (h = 108.0 μm, R=2.2h): almost no L1 peak; red (h = 1.08 μm, R=2.2h): dominant ZGV peak and significant L1 peak; green (h = 1.08 μm, R=22.1h): dominant L1 peak. (b) Measured response spectra, red (h = 1.08 μm, R2.2h): dominant ZVG peak and significant L1 peak green (h = 1.08 μm, R 10–15 h, and suppressed ZGV peak and dominant L1 peak.

Close modal

To obtain the decay behavior of the L1 resonance in an analytical form which can be fitted to experimental data, we modified the approximate solution given for shear resonances by Laurent et al.24 They expressed the surface normal displacement due to a given Lamb mode in cylinder coordinates uZ(r,t) as

uZ(r,t)=12π0+Cth(k)I(ω)B(k)J0(kr)eiωtkdk.
(1)

Here, J0 is the zero-order Bessel function, I(ω) is the spectral content of the laser excitation, B(k) is the 2D Fourier transform of the spatial profile of the excitation spot, Cth(k) is the thermo-elastic conversion coefficient into the normal displacement of a mode, and ω is to be understood as ω(k), given by the dispersion curve of the mode. For frequency domain LUS with a modulated source, I(ω) can be considered constant and B(k) for a Gaussian beam is proportional to exp(sk2). For thickness resonances, the integral is then approximated by expanding ω(k) and Cth(k) into Taylor series in the proximity of k = 0. For Cth(k), we adapt the theory by allowing for a constant term, yielding Cth(k)C0+C1k, as thickness-stretch modes, in contrast to thickness-shear modes, have (only) out-of-plane displacement at k = 0. We then obtain the approximate decay behavior of a thickness-stretch mode at the point of excitation as (see the supplementary material for a detailed derivation)

uZ(t)(At1+Bt1.5)eiωCte2παLt,
(2)

where ωC is the angular cut-off frequency of the corresponding mode and αL is the damping coefficient. The coefficients A and B account for the specific thermo-elastic coupling of the source into the mode and depend on the elastic properties of the plate, its thickness, and the size of the source. The result differs from what was found for thickness-shear modes24 by the additional t1 term. The approximation is valid for t>R2/(4D), where D is the curvature of the mode at ω=ωC, which can be calculated from the elastic properties and thickness of the plate.24,33

To validate Eq. (2), we fitted it to a simulated response with known attenuation, by minimizing the root-mean-squared-error (RMSE), leaving ωC,αL, A, and B as free parameters. The blue curve in Fig. 3(a) shows the simulated spectrum of the out-of-plane displacement |uZ| at the epicenter for an aluminum34 plate with h = 1.08 μm and a set longitudinal damping coefficient αL=2.95×106/s, which is equivalent to Qset=ωC(4παL)1=500 of the L1 resonance at 2.958 GHz. The response includes transient contributions from propagating modes, which are not accounted for by Eq. (2). To remove them, we calculate the time domain representation of the response as the inverse fast Fourier transform (FFT) of the complex spectrum, shown in Fig. 3(b). Here, the transient contributions concentrate in a short period after t = 0. We can thus remove them by omitting the response before t=t0, resulting in a red curve in Fig. 3(b). The FFT of this time-gated signal is shown in Fig. 3(a). The shape of the resonance peak is slightly affected by the processing, especially on its right slope. This indicates that the S2 mode above the mode cutoff [see Fig. 1(c)] is the major source for transient contributions, which are removed by the processing. After the time-gating, the simulated response is represented well by Eq. (2). Figure 3(c) shows the complex valued, time-gated response as (uZ) and (uZ) in red and the result of a least RMSE fit in green. The fitting function used is the FFT of Eq. (2), with fitting parameters A,B,ωC, and αL and taking into account the time-gate window. The example with a simulated SNR=20log[max(|uZ(f)|)/RMS(noise(f))] of 30 dB and a time-gate as shown in Fig. 3(b) results in a Q-value of Qfit=532. A simulation without added noise leads to QfitQset for t0>0.04 μs. We provide the supplementary material on the selection of the time-gate t0, the influence of noise, and the contribution of the two terms of Eq. (2). To summarize, simulations suggest that it is sufficient to choose t0R2/(4D) for Eq. (2) to provide a good approximation. For an SNR worse or comparable to what we can achieve experimentally, it was possible to choose a time-gate t0, large enough to eliminate transient contributions and at the same time small enough to preserve a reasonable SNR, resulting in Q-values within 10% of the set value.

FIG. 3.

Signal processing of the simulated response; (a) magnitude spectrum of raw (blue) and time gated (red) signal; (b) time domain representation of (a); and (c) complex spectrum of time gated response and fit.

FIG. 3.

Signal processing of the simulated response; (a) magnitude spectrum of raw (blue) and time gated (red) signal; (b) time domain representation of (a); and (c) complex spectrum of time gated response and fit.

Close modal

To achieve this SNR in our experiments, we averaged 40 frequency sweeps with 1500 points each. For each sweep, a baseline signal was recorded while automatically blocking the excitation beam and subtracted to correct for electromagnetic interferences. Depending on the thickness (thinner samples provided higher signals), we used a detection bandwidth of 100 or 300 Hz. This required approximately 25 min per spatial point. A measurement of the thinnest (h = 1.08 μm, f0=2.96 GHz) sample and the applied signal processing is shown in Figs. 4(a)–4(c). After signal processing, as described for the simulations above, we could still see slight discrepancies between fit and experimental data. These appear mainly for frequencies above the L1 resonance frequency. To investigate their origin, a spatial scan of the measurement position relative to the sample position was done. Therefore, an automated stage moved the samples in steps of 0.625 μm and a spectrum was recorded as described above in each position. The results are shown color-coded in Fig. 4(d), where y indicates the position along the short side of the sample (5 μm is approximately in the center and the edge is at 50 μm). The scan reveals features above the resonance frequency, which are periodic in y and change periodicity with f. Scans ranging over the whole sample show that these features are symmetric with respect to the center of the sample. Our hypothesis is that these are standing waves, originating from contributions of the S2 mode, which exist above the L1 resonance frequency [see Fig. 1(c)] and which are reflected from the sample's edges. In the simulations, which describe plates of infinite extension without an edge, these spurious modes are generated as well but appear only as transient contributions before they propagate away. While in the simulations they can be sufficiently removed by time-gating as described above, this has only a limited effect for the experiments, where they are reflected and resonate between the parallel edges of the sample. The resulting formation of additional guided wave modes35 explains the observed pattern. To reduce the influence of the standing waves on the fit, we restricted the fit range to the left flank of the peak, as indicated by the blue lines in Fig. 4(c). The spatial scan also shows a shift in the center frequency when moving towards the edge. A thickness variation caused by the production process is a plausible explanation for this shift.

FIG. 4.

Experimental results (a)–(d) for the 1.08 μm aluminum plate: (a) magnitude spectrum of raw (blue) and time gated (red) signals; (b) time domain representation of (a); (c) complex spectrum of the time gated response and fit; (d) spatial scan of the sample, showing standing waves above the L1 frequency; and (e) collected results for all samples.

FIG. 4.

Experimental results (a)–(d) for the 1.08 μm aluminum plate: (a) magnitude spectrum of raw (blue) and time gated (red) signals; (b) time domain representation of (a); (c) complex spectrum of the time gated response and fit; (d) spatial scan of the sample, showing standing waves above the L1 frequency; and (e) collected results for all samples.

Close modal

Applying the fitting routine with the limited fit range as indicated in Fig. 4(c) to all spectra of the spatial scan, we find that a broad region in the center converges to a Q-value around 500±10% for time gates with t0>0.04 μs for the shown sample (h = 1.08 μm and f0=2.96 GHz). We note that not all investigated samples showed a clear convergence towards one Q-value for increasing t0. We ascribe this to the interference of standing waves. We proceeded by averaging the fit results obtained for at least 30 different scan positions in each sample and a range of plausible time-gates t0R2/(4D). We used the inverse of the normalized residues from the least RMSE fit as weights for the averaging, thus prioritizing the best matching fit results. Figure 4(e) presents the found results for six frequencies corresponding to six samples with different thicknesses. The blue error bars indicate the weighted standard deviation of the experimental Q-values. To interpret the results, we compare them to Q-values calculated from the dispersion relation for longitudinal waves in the coupled thermoelasticity [derived from Eq. (10.12) in Achenbach2] shown as the red curve. For a homogeneous metallic sample, thermo-elastic damping is typically considered the main intrinsic damping-mechanism36 and provides an upper limit for the expected Q-values. The trend of the measured Q-values, which can be approximated by a power law Q(f)=Q0fn with n between 1.4 and 1.6 [colored areas in Fig. 4(e)], deviates from this theory, which predicts Q(f)f1. This indicates that other mechanisms contribute significantly to attenuation in the samples. Such could be extrinsic influences, like grain boundary scattering3–7 and surface roughness37 or other intrinsic damping channels mediated by electron-phonon or phonon-phonon interactions.36,38,39 We suggest a systematic study, e.g., with varying sample production process parameters, as an attempt to disentangle the contributions. This may also yield an optimized material behavior, benefiting the performance of micro-acoustic devices.

In conclusion, we presented a concept to evaluate frequency dependent material attenuation based on plate-resonances. We selectively excited and detected thickness-stretch resonances in the GHz-range using a spot size well above the plate thickness. It showed that the thermo-elastic coupling into thickness-stretch resonances is much stronger for micrometer-scale plate thicknesses than for thicknesses in the range of 0.1 mm and above. We derived an approximate theory for the decay behavior of longitudinal resonances, which we validated with simulations and fitted to the recorded responses. To obtain damping values for a range of frequencies, we produced a set of samples with the corresponding thicknesses. Within the present samples, reflections of spurious modes from the edges are influencing the results. For future investigations, we propose to increase the lateral dimensions of the investigated samples so that propagating modes reflected back from the edges of the sample decay below detectability. Another approach is to design samples without parallel edges to avoid the formation of standing waves. These options are currently tested in 3D simulations. While we aim to reduce the uncertainty of the measurement for the experimentally found Q-values, we are confident that the provided estimations for the values in the range of 1.5–3 GHz and their trend are of valuable information for the micro-acoustic-device industry. Our experiments showed that also the L2 resonance present at the double frequency can be detected. Its evaluation and also the inclusion of ZGV resonances,10 as well as different sample materials and thicknesses, are currently in progress.

See the supplementary material for the production process of the samples, the mathematical derivation of the temporal decay power law, and its validation with simulation data.

We kindly thank Claire Prada for helpful input on the decay power law for the thickness-stretch resonances and Gernot Fattinger for fruitful discussions throughout the project. This work was funded by the Austrian research funding association (FFG) under the scope of the COMET program within the research project “Photonic Sensing for Smarter Processes (PSSP)” (Contract No. 871974).

The data that support our findings are available from the corresponding author upon reasonable request.

1.
R.
Truell
,
C.
Elbaum
, and
B.
Chick
,
Ultrasonic Methods in Solid State Physics
, 1st ed. (
Academic Press
,
1969
), p.
180
.
2.
J. D.
Achenbach
,
Wave Propagation in Elastic Solids
(
Elsevier Science
,
1993
), p.
394
.
3.
X.-G.
Zhang
,
W. A.
Simpson
,
J. M.
Vitek
,
D. J.
Barnard
,
L. J.
Tweed
, and
J.
Foley
,
J. Acoust. Soc. Am.
116
,
109
(
2004
).
4.
C. M.
Kube
and
J. A.
Turner
,
J. Acoust. Soc. Am.
137
,
EL476
(
2015
).
5.
M.
Ryzy
,
T.
Grabec
,
P.
Sedlák
, and
I. A.
Veres
,
J. Acoust. Soc. Am.
143
,
219
(
2018
).
6.
M.
Ryzy
,
T.
Grabec
,
J. A.
Österreicher
,
M.
Hettich
, and
I. A.
Veres
,
AIP Adv.
8
,
125019
(
2018
).
7.
B.
Lehr
,
H.
Ulrich
, and
O.
Weis
,
Z. Phys. B
48
,
23
(
1982
).
8.
Q.
Xie
,
S.
Mezil
,
P. H.
Otsuka
,
M.
Tomoda
,
J.
Laurent
,
O.
Matsuda
,
Z.
Shen
, and
O. B.
Wright
,
Nat. Commun.
10
,
1
(
2019
).
9.
S.
Bramhavar
,
B.
Pouet
, and
T. W.
Murray
,
Appl. Phys. Lett.
94
,
114102
(
2009
).
10.
T.
Berer
,
C.
Grunsteidl
,
R.
Rothemund
,
S.
Kreuzer
,
M.
Ryzy
,
M.
Hettich
, and
I. A.
Veres
, in
IEEE International Ultrasonics Symposium, IUS
(
2019
).
11.
J. D.
Aussel
and
J. P.
Monchalin
,
J. Appl. Phys.
65
,
2918
(
1989
).
12.
J.
Cuffe
,
O.
Ristow
,
E.
Chávez
,
A.
Shchepetov
,
P.-O.
Chapuis
,
F.
Alzina
,
M.
Hettich
,
M.
Prunnila
,
J.
Ahopelto
,
T.
Dekorsy
, and
C. M.
Sotomayor Torres
,
Phys. Rev. Lett.
110
,
095503
(
2013
).
13.
C. J.
Morath
and
H. J.
Maris
,
Phys. Rev. B
54
,
203
(
1996
).
14.
W.
Arnold
,
B.
Betz
, and
B.
Hoffmann
,
Appl. Phys. Lett.
47
,
672
(
1985
).
15.
A.
Moreau
,
D.
Lévesque
,
M.
Lord
,
M.
Dubois
,
J. P.
Monchalin
,
C.
Padioleau
, and
J. F.
Bussière
,
Ultrasonics
40
,
1047
(
2002
).
16.
C.
Prada
,
O.
Balogun
, and
T. W.
Murray
,
Appl. Phys. Lett.
87
,
194109
(
2005
).
17.
D.
Clorennec
,
C.
Prada
,
D.
Royer
, and
T. W.
Murray
,
Appl. Phys. Lett.
89
,
024101
(
2006
).
18.
D.
Clorennec
,
C.
Prada
, and
D.
Royer
,
J. Appl. Phys.
101
,
034908
(
2007
).
19.
C.
Grünsteidl
,
T. W.
Murray
,
T.
Berer
, and
I. A.
Veres
,
Ultrasonics
65
,
1
(
2016
).
20.
S.
Raetz
,
J.
Laurent
,
T.
Dehoux
,
D.
Royer
,
B.
Audoin
, and
C.
Prada
,
J. Acoust. Soc. Am.
138
,
3522
(
2015
).
21.
C.
Grünsteidl
,
T.
Berer
,
M.
Hettich
, and
I.
Veres
,
Appl. Phys. Lett.
112
,
251905
(
2018
).
22.
J. L.
Rose
,
Ultrasonic Waves in Solid Media
, 1st ed. (
Cambridge University Press
,
Cambridge, UK
,
1999
), p.
123
.
23.
C.
Prada
,
D.
Clorennec
, and
D.
Royer
,
Wave Motion
45
,
723
(
2008
).
24.
J.
Laurent
,
D.
Royer
, and
C.
Prada
,
Wave Motion
51
,
1011
(
2014
).
25.
C. M.
Grünsteidl
,
I. A.
Veres
, and
T. W.
Murray
,
J. Acoust. Soc. Am.
138
,
242
(
2015
).
26.
To achieve the GHz-range bandwidth, we selected an according photo-detector (Alphalas UDP-50-SP), amplifier (Miteq AFS4-00100600-13-10P-4) and used an 8.5 GHz vector-network-analyzer (Rhode&Schwarz ZNB8), replacing the lock-in-amplifier and the signal generator from previous setups.
27.
C. B.
Scruby
and
L. E.
Drain
,
Laser Ultrasonics Techniques and Applications
(
CRC Press, Taylor and Francis Group
,
1990
), p.
76
.
28.
C.
Grünsteidl
, “
Effects and applications associated with zero group velocity Lamb waves
,” Doctoral thesis (
Johannes Kepler Universität
,
2017
).
29.
T. W.
Murray
and
O.
Balogun
,
Appl. Phys. Lett.
85
,
2974
(
2004
).
30.
C.
Grünsteidl
,
I. A.
Veres
,
J.
Roither
,
P.
Burgholzer
,
T. W.
Murray
, and
T.
Berer
,
Appl. Phys. Lett.
102
,
011103
(
2013
).
31.
F.
Bruno
,
J.
Laurent
,
P.
Jehanno
,
D.
Royer
, and
C.
Prada
,
J. Acoust. Soc. Am.
140
,
2829
(
2016
).
32.
O.
Balogun
,
T. W.
Murray
, and
C.
Prada
,
J. Appl. Phys.
102
,
064914
(
2007
).
33.
A. L.
Shuvalov
and
O.
Poncelet
,
Int. J. Solids Struct.
45
,
3430
(
2008
).
34.
Thermal properties of simulated aluminum are: Thermal expansion coefficient aT=23×106/K, thermal conductivity KT=160 W/mK, heat capacity CT=900 J/kgK.
35.
J.
Laurent
,
D.
Royer
, and
C.
Prada
,
J. Acoust. Soc. Am.
147
,
1302
(
2020
).
36.
R.
Tabrizian
,
M.
Rais-Zadeh
, and
F.
Ayazi
, in
The 15th International Conference on Solid-state Sensors, Actuators and Microsystems (Transducers 2009)
(
2009
), Vol.
2131
.
37.
A. A.
Maznev
,
Phys. Rev. B
91
,
134306
(
2015
).
38.
S. A.
Chandorkar
,
M.
Agarwal
,
R.
Melamud
,
R. N.
Candler
,
K. E.
Goodson
, and
T. W.
Kenny
, in
Proceedings of the IEEE International Conference on Micro Electro Mechanical Systems (MEMS)
(
2008
), p.
74
.
39.
E.
Chávez-Ángel
,
R. A.
Zarate
,
J.
Gomis-Bresco
,
F.
Alzina
, and
C. M.
Sotomayor Torres
,
Semicond. Sci. Technol.
29
,
124010
(
2014
).

Supplementary Material