ILAMP (Interdisciplinary Laboratories for Advanced Materials Physics),
Università Cattolica del Sacro Cuore, IT-25133 Brescia, Italy Department of
Physics and Astronomy, KU Leuven, B-3001 Leuven, Belgium ILAMP
(Interdisciplinary Laboratories for Advanced Materials Physics), Università
Cattolica del Sacro Cuore, IT-25133 Brescia, Italy CNR-INO (National Institute
of Optics), via Branze 45, IT-25123 Brescia, Italy Dipartimento di Fisica,
Università degli Studi di Trieste, Italy International Faculty, University of
Cologne, Albertus-Magnus-Platz, 50923 Cologne, Germany

# Impact of a MoS2 monolayer on the nanoscale thermoelastic response of
silicon heterostructures

Davide Soranzio Institute for Quantum Electronics, Eidgenössische Technische
Hochschule (ETH) Zürich, CH-8093 Zurich, Switzerland<EMAIL_ADDRESS>Denny Puntel Dipartimento di Fisica, Università degli Studi di Trieste, Italy
Manuel Tuniz Dipartimento di Fisica, Università degli Studi di Trieste, Italy
Paulina E. Majchrzak Department of Physics and Astronomy, Interdisciplinary
Nanoscience Center (iNANO), Aarhus University, 8000 Aarhus C, Denmark
Alessandra Milloch Department of Mathematics and Physics, Università Cattolica
del Sacro Cuore, IT-25133 Brescia, Italy Nicholas M. Olsen Department of
Chemistry, Columbia University, New York, NY, USA Wibke Bronsch Elettra -
Sincrotrone Trieste S.C.p.A., Strada Statale 14, km 163.5, Trieste, Italy
Bjarke S. Jessen Department of Physics, Columbia University, New York, NY, USA
Danny Fainozzi Elettra - Sincrotrone Trieste S.C.p.A., Strada Statale 14, km
163.5, Trieste, Italy Jacopo S. Pelli Cresi Elettra - Sincrotrone Trieste
S.C.p.A., Strada Statale 14, km 163.5, Trieste, Italy Dario De Angelis Elettra
- Sincrotrone Trieste S.C.p.A., Strada Statale 14, km 163.5, Trieste, Italy
Laura Foglia Elettra - Sincrotrone Trieste S.C.p.A., Strada Statale 14, km
163.5, Trieste, Italy Riccardo Mincigrucci Elettra - Sincrotrone Trieste
S.C.p.A., Strada Statale 14, km 163.5, Trieste, Italy Xiaoyang Zhu Department
of Chemistry, Columbia University, New York, NY, USA Cory R. Dean Department
of Physics, Columbia University, New York, NY, USA Søren Ulstrup Department of
Physics and Astronomy, Interdisciplinary Nanoscience Center (iNANO), Aarhus
University, 8000 Aarhus C, Denmark Francesco Banfi Université de Lyon, CNRS,
Université Claude Bernard Lyon 1, Institut Lumière Matière, F-69622
Villeurbanne, France Claudio Giannetti Department of Mathematics and Physics,
Università Cattolica del Sacro Cuore, IT-25133 Brescia, Italy Fulvio
Parmigiani Elettra - Sincrotrone Trieste S.C.p.A., Strada Statale 14, km
163.5, Trieste, Italy Filippo Bencivenga Elettra - Sincrotrone Trieste
S.C.p.A., Strada Statale 14, km 163.5, Trieste, Italy Federico Cilento Elettra
- Sincrotrone Trieste S.C.p.A., Strada Statale 14, km 163.5, Trieste, Italy
<EMAIL_ADDRESS>
Abstract

Understanding the thermoelastic response of a nanostructure is crucial for the
choice of materials and interfaces in electronic devices with improved and
tailored transport properties, at the length scales of the present technology.
Here we show how the deposition of a MoS2 monolayer can strongly modify the
nanoscale thermoelastic dynamics of silicon substrates close to their
interface. We achieve this result by creating a transient grating with extreme
ultraviolet light, using ultrashort free-electron laser pulses, whose
$\approx$84 nm period is comparable to the size of elements typically used in
nanodevices, such as electric contacts and nanowires. The thermoelastic
response, featured by coherent acoustic waves and an incoherent relaxation, is
tangibly modified by the presence of monolayer MoS2. Namely, we observed a
major reduction of the amplitude of the surface mode, which is almost
suppressed, while the longitudinal mode is basically unperturbed, aside from a
faster decay of the acoustic modulations. We interpret this behavior as a
selective modification of the surface elasticity and we discuss the conditions
to observe such effect, which might be of immediate relevance for the design
of Si-based nanoscale devices.

Transition metal dichalcogenides (TMDs) are a class of materials composed of
atomic layers held together by van der Waals interactions, much weaker than
the intralayer covalent bonds, giving them a marked two-dimen- sional
character. This feature allows to obtain controlled thicknesses down to the
single layers, useful for tuning several physical properties, such as the
electronic band gap, the vibrational levels and the excitons, and for their
implementation in thin nanodevices, e.g., transistors, photodetectors and
electroluminescent devices 1, 2, 3, 4. In this respect, a key point is the
interaction between TMDs and substrates, which has been shown to critically
impact the heterostructure properties. Hence, the choice of the substrate
constitutes a fundamental aspect for the application of TMDs heterostructures
in electronic circuits 5, 6, 7, 8. To design functional devices, the
thermoelastic response of the structure, i.e., how fast can heat be
dissipated, together with its elastic properties, which are also relevant for
technological applications 9, 10, are fundamental aspects to be considered.

In this work, we focus on the dynamics over the typical length-scales of
components used in nanodevices, e.g., electric contacts and nanowires 11, 12.
In particular, we study the nanoscale thermoelastic response of a TMD
monolayer (ML) deposited on top of a silicon substrate. The selected TMD
material is molybdenum disulfide (MoS2), which presents an interlayer
separation of 0.65 nm and a $\approx$1.9 eV direct band gap at the ML limit
13. To investigate the thermoelastic dynamics, we used the transient grating
(TG) technique 14. Specifically, nanoscale TGs were generated on the sample
using free electron laser (FEL) extreme-ultraviolet (EUV) femtosecond pulses,
to attain a $\approx$84 nm grating period. We studied the TG dynamics in three
systems: a blank Si membrane, a ML MoS2 transferred on a Si membrane and a ML
MoS2 grown on a Si wafer. The results showed a marked difference in the
surface thermoelastic response of Si when ML MoS2 is added. While two acoustic
modes, a superficial Rayleigh-like and a longitudinal, were detected for the
blank membrane, the longitudinal mode dominates for the MoS2-covered samples
alongside with a faster decay of the modulations. We ascribe this fact to the
different atomic motion involved, which is elliptical and localized close to
the interface for the surface mode and parallel to the interface for the
longitudinal mode. Underneath the acoustic modulations, we observed a slower
decay of the non-oscillating background, usually assigned to thermal transport
15, for the heterostructures compared to the blank substrate.

##  1 Results and discussion

A schematic of the samples is shown in Fig. 1(a). A blank Si membrane was used
as a reference (sample #1). On top of a nominally identical membrane, a large-
area monolayer of MoS2 was prepared using the gold tape method (sample #2) 16.
For comparison to the membrane systems, we also studied a CVD-grown ML MoS2,
deposited on a Si wafer (sample #3). More details regarding the sample
preparation are reported in the Methods section.

The MoS2 coverage was tested using micro-Raman, taking advantage of the
relation between the frequencies of the MoS2 $E^{1}_{2g}$ and $A_{1g}$ near-
zone-center phonon modes and number of layers 1. Sample scans are reported in
Fig. 1(b), with the ML-covered regions from sample #2 and #3 showing a much
smaller ($\approx$19 cm-1) frequency separation of the modes, when compared to
a thick multilayer, bulk-like island found on the Si frame ($\approx$25 cm-1),
which is consistent with the literature. A systematic characterization of the
samples is reported in the Supplemental Material 17. Using the bulk MoS2 film
island as a reference, we conclude that the values found for both sample #2
and #3 are compatible with a ML coverage.

Figure 1: (a) Samples studied during the TG experiments: #1 blank Si membrane,
#2 MoS2/Si membrane and #3 MoS2/Si wafer. (b) Selected micro-Raman spectra
showing the $E^{1}_{2g}$ and $A_{1g}$ phonon peaks from the investigated
samples: MoS2/Si membrane, MoS2/Si wafer and a MoS2 bulk island. The Raman
traces were rescaled and vertically shifted for clarity. The blue lines
correspond to the experimental data, while the red-dotted ones to the best
fit. (c) Scheme of the TG setup at the EIS-TIMER beamline at FERMI.  Figure 2:
(a) TG signal from the blank Si membrane (sample #1) at 0.34 mJ/cm2 fluence;
the inset shows the Fourier transform of the signal after the subtraction of
an exponentially-decaying background for the $\approx$160 ps window. (b) TG
signal from the MoS2/Si membrane (sample #2) at 0.23 mJ/cm2 fluence compared
to the one from (a); the inset shows the Fourier transform of the two signals
after the subtraction of an exponentially-decaying background for the
$\approx$60 ps window; two integration region are marked (see text) (c) TG
signal collected at different positions compared to (b) on the membrane of
sample #2 under 0.25 mJ/cm2 fluence.

Fig. 1(c) illustrates the EUV TG experimental geometry. Two symmetric pump
beams with wavelength $\lambda_{pu}=39.9$ nm and time duration $<$ 100 fs
cross at the sample with an angle $2\theta$=27.6∘ leading to a TG with a
periodicity $\Pi=2\pi/k_{TG}$=83.6 nm, where $k_{TG}=4\pi
sin(\theta)/\lambda_{pu}$ is the modulus of the TG vector. The probe beam was
set at $\lambda_{pr}=13.3$ nm. More details regarding the EUV TG setup are
provided in the Methods section. All the measurements were performed in
reflection geometry, in a high vacuum and at room temperature. The diffracted
signal was recorded using a CCD camera.

Fig. 2(a) shows the time-resolved TG signal for the blank Si membrane (sample
#1). Before the temporal overlap between pump and probe, occurring at $t$=0,
no signal is detected, while for $t>0$ a transient diffracted signal appears.
It is featured by modulations on top of the decay of the average signal, which
is typical for a thermoelastic response.

To analyze the TG signal, we model its time-resolved intensity using the
expression 15

$I(t)=\theta(t)\cdot\\\
\left|A_{th}e^{-t/\tau_{th}}+\sum^{n}_{i=1}A_{i}e^{-t/\tau_{i}}\cos\left(2\pi
f_{i}t+\phi_{i}\right)\right|^{2}$ (1)

where $\theta(t)$ is the Heaviside function, $A_{th}$ and $\tau_{th}$ are the
amplitude and decay time usually assigned to the thermal component, $A_{i}$,
$f_{i}$, $\tau_{i}$ and $\phi_{i}$ are the amplitude, frequency, time decay
constant and phase of the i-th mode. Two modes are sufficient to describe the
TG signal from sample #1. In fact, the square in the formula leads, when
expanded, to a series of exponentially-decaying terms oscillating at
frequencies 0, $f_{1}$, $f_{2}$, $2f_{1}$, $2f_{2}$, $f_{1}+f_{2}$ and
$f_{1}-f_{2}$. The non-null frequency contributions can be visualized by
fitting the experimental data to Eq. 1 with the oscillatory amplitudes $A_{i}$
set to zero and performing the discrete Fourier transform on the residual
signal (inset of Fig. 2(a)). The result is consistent with two modes with
overtones, corresponding to the sums and differences of two frequencies
$f_{1}$ and $f_{2}$, such that $f_{1}\approx 102$ GHz, $f_{2}\approx 60$ GHz,
$2f_{1}\approx 204$ GHz, $2f_{2}\approx 120$ GHz, $f_{1}+f_{2}\approx 162$ GHz
and $f_{1}-f_{2}\approx 42$ GHz, with the last one appearing as the left
shoulder of the most prominent peak at $f_{2}\approx 60$ GHz. By fitting the
data using the full Eq. 1, we obtain the exact frequencies of the two modes,
$f_{1}=(105\pm 1)$ GHz and $f_{2}=(58.2\pm 0.4)$ GHz. Moreover, the decay time
is found to be $\tau_{th}=(39\pm 2)$ ps.

Figure 3: Schematic representation of the displacements for the (a)
longitudinal (b) Rayleigh acoustic waves in silicon. Their amplitude was
arbitrarily set by a common factor to ease visualization; in the colorbar, the
displacements were normalized to the maximum one with respect to the
equilibrium positions. Figure 4: TG signal from sample #3 (a) Fluence
dependence of TG signal from sample #3. (b) Comparison between the normalized
TG signals from sample #2 (0.23 mJ/cm2) and sample #3 (0.25 mJ/cm2). (c)
Amplitude and (d) time decay constant of the non-oscillating component, as
derived by the best fit of EUV TG waveforms from panel (a) to Eq. 1.

We can rationalize the presence of these two modes as Lamb waves, namely,
acoustic waves of the membrane 18, 19. In TG experiments, their momentum is
set by $k_{TG}$, while the propagation speed is related to the frequency-
momentum dispersion relation of the material. Using the frequencies derived
from the TG signal, it is possible to obtain the corresponding phase
velocities with $v_{p,i}=f_{i}\Pi$. For sample #1, the detected modes
correspond to $v_{p,1}=(8.81\pm 0.09)$ km/s and $v_{p,2}=(4.87\pm 0.03)$ km/s.
These can be compared to the predicted phase velocity dispersion for the
200-µm-thick Si membrane, through the transverse resonance method 18, 20, and
using the longitudinal and shear velocities reported in Farnell and Adler 21.
The resulting phase velocities depend on the product $\rho=k_{TG}b$, where, in
the present case, $k_{TG}$=0.075 nm-1 and $b=200$ nm is the membrane
thickness; hence $\rho\approx 15$. At this $\rho$ value, two modes with phase
velocities 8.89 km/s and 4.90 km/s, close to the longitudinal and transverse
ones for Si, are predicted, compatible with the observed ones. The fitting
procedure also provides the acoustic decay constants, which result
$\tau_{1}=(80\pm 50)$ ps and $\tau_{2}\approx 470$ ps. Both values are either
comparable or larger than the investigated delay range, as shown in Fig. 2(a),
hence a more accurate estimation would require further dedicated measurements.

Similar results have been previously reported for optical and EUV TG
experiments in other materials for $\rho\gg 1$ 22, 23, 24, 25, 26. The
acoustic modes are commonly identified as the ‘surface-skimming’ longitudinal
wave (SSLW) and, in the limit of a semi-infinite substrate ($\rho\to\infty$),
the Rayleigh surface acoustic wave (RSAW). As observed in our measurements,
the SSLW at $f_{1}$=105 GHz is only visible in the first $\approx$15 ps in the
TG signal, since it behaves like a leaky wave which rapidly decays in the bulk
22. The RSAW decays on a longer timescale and involves elliptical
displacements, i.e., combined longitudinal and transversal motion. These are
approximately localized at the surface within half a wavelength 27, 19. A
depiction of the two waves is given in Fig. 3, based on the derivation of
their displacements with the method of potentials 19; more details are
provided in the Supplementary Information 17. The accurate prediction of the
wave and heat propagation is complex and depends on many experimental details,
such as the energy distribution in time and space. Nevertheless, in the
reflection configuration used in the experiments, one expects the dominant
contribution to the TG signal to arise from a coherent surface displacement,
i.e., from the RSAW 28, 29, 22, 30.

In Fig. 2(b) we show the time-resolved TG signal for the MoS2/Si membrane
heterostructure (sample #2) compared to the blank membrane (sample #1).
Differently from the blank case, here we observe an oscillatory signal in the
first $\approx$15 ps only, with two prominent peaks. By performing the
discrete Fourier transform after subtracting the non-oscillatory signal from
the two traces as for panel (a), we observe a clear difference between sample
#1 and sample #2 as shown in the inset in panel (b). In fact, if we integrate
the spectral amplitude around the two main mode frequencies ($f_{1}$, 90-120
GHz) and ($f_{2}$, 45-75 GHz), we obtain that their ratio is I1/I2$\approx$0.6
for sample #1 and $\approx$1.9 for sample #2, pointing to a marked difference
in the relative contribution of the two modes. Afterwards, the signal
gradually decays with minor modulations on a longer timescale with respect to
the blank substrate (Fig. 2(a),(c)). Repeated scans at different sample
positions gave an analogous response.

Fig. 4(a) shows a fluence dependence of the TG signal collected on the MoS2/Si
wafer (sample #3). Analogously to sample #2, we observe a response mainly
localized in the first picoseconds with two main oscillation periods,
resembling the SSLW detected in sample #1, followed by lower frequency
modulations that emerge only for the highest investigated fluences, similar to
the ones associated to the RSAW observed for sample #1. The TG signal acquired
at different positions on the sample showed analogous features, albeit with
some amplitude variability possibly connected to the local MoS2 coverage (see
Supplementary Information 17). A comparison between the response of samples #2
and #3 under similar excitation conditions is reported in Fig. 4(b), showing
similar features in the dynamics.

Considering the values of the attenuation length for Si and MoS2, reported in
the Methods section (at 39.9 nm, $\approx$237 nm and $\approx$40 nm
respectively), it is evident that a MoS2 ML cannot significantly alter the
amount of energy deposited in the Si substrates. On the other hand, the
monolayer/substrate interaction leads to a thermal resistance and changes in
the dynamical heat distribution and elasticity at the interface of MoS2/Si
heterostructures; these properties are also influenced by the surface
nanogroove of the substrate which is connected dynamically to the
thermoelastic response 31, 32. Therefore, one may expect surface modes like
the RSAW, involving both longitudinal and transverse displacements 19, to be
more affected by the interaction with the MoS2 ML rather than the longitudinal
wave that leaks into the bulk. This reasoning is consistent with the
experimental results reported in Figs. 2(b),(c) and 4(a), where a main
oscillatory contribution with frequency close to the longitudinal mode
dominates the TG signal of the samples #2 and #3, where a ML MoS2 is present.

To give a more quantitative view, we fitted the data from sample #2 and #3
using Eq. 1. Besides the TG signal collected from sample #3 at the highest
fluences, the resolved oscillations are limited in time to about two periods
of the longitudinal mode. This does not give enough data to effectively
disentangle the contributions at distinct frequencies due to the correlation
among the parameters, especially for the lower fluences. Nonetheless, it is
possible to give good estimates keeping some of the parameters fixed. For
sample #2, fixing the frequencies and phases of the two modes from the fit to
the sample #1 TG signal (Fig. 2(a)), we obtained an average non-oscillating
background decay constant $\tau_{th}=(170\pm 10)$ ps over the different
positions on the sample at 0.25 mJ/cm2 (Fig. 2(c)). For sample #3, we first
model the response at 0.56 mJ/cm2 (Fig. 4(a)), where both modes emerge,
keeping all the parameters free. The resulting frequencies are $f_{1}=(113\pm
5)$ GHz and $f_{2}=(55\pm 1)$ GHz, which are connected to the velocities
$v_{1}=(9.4\pm 0.4)$ km/s and $v_{2}=(4.62\pm 0.09)$ km/s, close to the ones
of the membrane. Regarding the decay constant of the acoustic modes, we
obtained $\tau_{1}=(5\pm 1)$ ps and $\tau_{2}=(70\pm 40)$ ps. We then fix the
frequencies, phases and time decay constants of the acoustic modes to extract
the fluence dependence of the amplitude $A_{th}$ and time decay constant
$\tau_{th}$, which we report in Fig. 4(c),(d). We omit the fit results for the
lowest fluence as the time decay parameter is heavily influenced by the lower
signal-to-noise ratio of this trace. While $A_{th}$ increases linearly in the
explored fluence range, $\tau_{th}$ settles around $\approx$100 ps for most of
the explored range. The higher value $\tau_{th}=(220\pm 60)$ ps from sample #3
at 0.23 mJ/cm2 may appear as an outlier, however it is compatible with the
value derived from the sample #2 data under close excitation conditions
($\tau_{th}=(170\pm 10)$ ps at 0.25 mJ/cm2).

Figure 5: Contribution of the acoustic modes to the TG signal illustrated by
comparing the experimental data (Exp.) to the best-fit curve using Eq. 1 (Best
fit) and by selectively removing their contribution from the fit result (only
SSLW, only RSAW and no osc.) for samples (a) #1 (b) #2 (c) #3.

To illustrate how the effects of the two acoustic modes combine differently
between the blank Si membrane and the MoS2 heterostructures, in Fig. 5 we
present examples from the three samples: (a) #1 at 0.34 mJ/cm2, (b) #2 at 0.23
mJ/cm2 and (c) #3 at 0.56 mJ/cm2. Here we remove from the fit (indicated with
‘Best fit’), obtained using Eq. 1, the contribution of one (‘Only SSLW’ or
‘Only RSAW’) or both (‘No osc.’) of the two acoustic modes, by setting their
amplitude to zero. It is clear that the double-peak structure found in the
MoS2 heterostructures originates from the superposition between longitudinal
and Rayleigh oscillations, where the relative height of the first two peaks is
dictated by the combination of amplitude and phase of the two modes, more
rapidly damped than the sample #1 case.

The observation of the influence from the ML MoS2 on the TG signal is favoured
by the choice of the substrate. We see from Fig. 4(a) that the amplitudes of
the acoustic modes increase with the pump fluence, that determines larger
displacements from the atomic equilibrium positions. The attenuation length of
our pump beams is $\approx$237 nm, comparable to the membrane thickness
$b\approx 200$ nm for sample #2. In another commonly employed substrate for
TMDs such as sapphire, the attenuation length is just $\approx$13 nm, about 18
times smaller 17. This gives a much higher excitation energy density than in
the case of Si, potentially giving much larger displacements due to the RSAW
mode, which are localized within a depth of half of the mode wavelength in the
material (Fig. 3(b)). For fluences comparable to the ones employed for the
data reported in Figs. 2 and 4, the RSAW in the TG signal dominates the
response also in a ML WS2/Sapphire heterostructure. We provide an example of
this in the Supplemental Material 17.

Finally, we discuss the decay of the non-oscillatory component of the TG
signal. On sample #1, we obtained $\tau_{th}=(39\pm 2)$ ps. This value is
substantially slower than what is expected by the standard thermal diffusive
theory. Using the Si thermal diffusivity $\alpha_{S}\approx 0.055$ µm2/nS from
Johnson et al. 33 and our grating vector $k_{TG}=0.075$ nm-1, the thermal
diffusion theory predicts a decay $\tau_{Si}=1/\alpha_{s}k^{2}_{TG}\approx$3.2
ps, about one order of magnitude smaller than the value from our experiment.
This deviation was previously described as a consequence of ballistic
transport, occurring for $k_{TG}\Lambda_{mfp}\gg 1$, where $k_{TG}$ is the TG
vector and $\Lambda_{mfp}$=0.5-1 µm is the median phonon mean free path 33, 34
.

With the addition of ML MoS2, we observed a marked increase in the non-
oscillatory time decay in samples #2 and #3. However, the decay constant found
for the heterostructure is almost as large or higher than the investigated
temporal windows for sample #2 and #3 (Fig. 2(b),(c) and 4(a)). Therefore, a
further dedicated set of measurements is required for an accurate explanation
of this result. Based on the considerations for Si, it might be important to
take into consideration a model that includes non-diffusive thermal transport,
the interfacial thermal resistance in the MoS2 and Si layers 5, 32 and
possibly the influence of charge recombination at the MoS2/Si interface 35.
Furthermore, as we have shown in our work, the amplitude and time decay of the
oscillations of the acoustic modes are markedly affected by the deposition of
ML MoS2, suggesting a possibly less effective contribution to the thermal
transport.

##  2 Conclusion

In this work we used nanoscale EUV transient gratings to comparatively study
the nanoscale thermoelastic response of a thin Si membrane and a Si wafer,
both covered with ML MoS2, and of a blank, thin Si membrane. The results show
how a monolayer of a TMD material, in our case MoS2, can dramatically modify
the response of the Si substrate close to its surface. In particular, ML MoS2
leads to the almost complete suppression of the surface acoustic wave, which
involves elliptical motion localized in proximity of the surface, for
comparable excitation fluences. Differently, the longitudinal wave, which does
not involve a motion perpendicular to the surface, appears to be less
affected, albeit more quickly damped as well. The two Si substrates used, a
membrane and a wafer, gave a qualitatively similar behavior. Finally, we
observed a slower decay of the non-oscillatory response in the
heterostructure, when compared to the blank substrate. This effect is possibly
connected to the impact of such acoustic modes on the transport properties of
the system at the nanoscale. This work extends the time-domain investigations
of the role of Van der Waals materials on the acoustic response of
heterostructures, beyond the case of coherently excited thin layer breathing
modes investigated, e.g. by time-resolved Brillouin spectroscopy 36.

##  3 Methods

###  3.1 Samples

Our study focused on samples based on a 200-nm-thick, 1.6x1.6-mm2-large boron-
doped Si membrane, commercially produced by Norcada 37, nominally having $<$ 1
nm rms surface roughness on both front and back sides of the membrane,
supported by a 200-µm-thick Si frame. The three samples are depicted in Fig.
1(a). One blank Si membrane was used as a reference (sample #1). On top of a
nominally identical membrane from the same manufacturer, a large-area
monolayer of MoS2 was prepared using the gold tape method 16 (sample #2) at
Columbia University. Additionally, we studied a CVD-grown MoS2 monolayer,
deposited on a 0.52-mm-thick Si wafer, commercially available from SixCarbon
Technology (sample #3).

###  3.2 Micro-Raman characterization

The uniformity and thickness of the MoS2 films were tested with a micro-Raman
Renishaw inVia instrument using a continuous-wave 532 nm laser at the I-LAMP
laboratories in Brescia (Italy). Measurements were acquired under an incident
power of 5 mW for the samples with a deposited MoS2 monolayer and 0.5 mW for
the blank substrates. At focus, under 100x magnification, the spot size is of
the order of a few microns. All the measurements were performed in air.

###  3.3 EUV transient-grating

The EUV gratings were generated at the EIS-TIMER beamline 38, 39, 15 using the
radiation generated by the double stage FEL source (FEL2) available at the
FERMI free-electron laser radiation 40. Fig. 1(c) illustrates the EUV TG
experimental geometry. Two symmetric pump beams with wavelength
$\lambda_{pu}=39.9$ nm and time duration $<$ 100 fs, generated by the first
stage of the FEL, are crossed at the sample with an angle $2\theta$=27.6∘
generating a TG with a periodicity $\Pi=2\pi/k_{TG}$=83.6 nm, where
$k_{TG}=4\pi sin(\theta)/\lambda_{pu}=0.075$ nm-1 is the modulus of the TG
vector. As a probe, we used the output of the second stage of the FEL, which
was set at $\lambda_{pr}=13.3$ nm.

The spot sizes, full width at half maximum (FWHM), of the two pump beams were
(380x290) µm2 and (470x250) µm2. The probe beam at the sample had a spot size
of (450x330) µm2. Unlike a typical pump-probe experiment, in a TG measurement
the recorded signal originates only from the probe beam fraction which is
diffracted by the transient grating. Thus, it only results from the region
where the three beams are superimposed. All the measurements were performed in
reflection geometry, in high vacuum and at room temperature. The diffracted
signal was recorded using a CCD camera. The fluence values reported in the
text refer to the sum of the incident fluences of the two pumps, whose pulse
energies are approximately equal. The FEL intensity at the sample was varied
by using a nitrogen gas cell, which efficiently attenuates the radiation at
$\lambda_{pu}$ while it marginally affects the transmission of the probe beam.
Typically, the acquisition of the EUV-TG signal takes about 1-2 minutes per
delay point, leading to 1-2 hours for a full trace and a good signal-to-noise
ratio. An estimate of the uncertainty over the TG signal values at EIS-TIMER
beamline was discussed by Bencivenga et al. 15. The spatial and temporal
overlap between the two pump and the probe beams were checked separately using
a reference IR laser and performing FEL-pump/IR-probe transient transmissivity
experiments on a YAG reference sample.

The attenuation lengths for the incident FEL pumps in MoS2 and Si can be
derived from the tabulated atomic data found in Henke et al.41. We obtain
$\approx$40 nm and $\approx$237 nm, respectively, while these values are
$\approx$205 nm and $\approx$583 nm at $\lambda_{pr}=13.3$ nm. An extended
wavelength dependence and comparison with other common substrates for MoS2 can
be found in the Supplemental Material 17.

A.M. and C.G. acknowledge financial support from MIUR through the PRIN 2015
(Prot. 2015C5SEJJ001) and PRIN 2017 (Prot. 20172H2SC4_005) programs and from
the European Union - Next Generation EU through the MUR-PRIN2022 (Prot.
20228YCYY7) program.

C.G. acknowledges support from Università Cattolica del Sacro Cuore through
D.1, D.2.2 and D.3.1 grants.

Sample fabrication of sample #2 was supported by the Columbia Nano Initiative
and Columbia Materials Science and Engineering Research Center (MRSEC) through
NSF grant DMR-2011738.

P.M and S.U. acknowledge funding from the Danish Council for Independent
Research, Natural Sciences under the Sapere Aude program (Grant No. DFF-
9064-00057B), VILLUM FONDEN under the Centre of Excellence for Dirac Materials
(grant 11744), and from the Novo Nordisk Foundation (Project Grant
NNF22OC0079960).

Supplementary Information on Impact of a MoS2 monolayer on the nanoscale
thermoelastic response of silicon heterostructures

##  4 Micro-Raman characterization

In the main text, selected scans of the micro-Raman characterization for the
Stokes scattering signal were reported in Fig. 1(b). The spectral window was
centered around two peaks corresponding to the $E^{1}_{2g}$ and $A_{1g}$ near-
zone center phonon modes of the MoS2 film with frequencies at $\approx$385
cm-1 and $\approx$404 cm-1 respectively. In Fig. S1 we report, measured at the
same sample positions, the peaks corresponding to the (degenerate) optical
phonon mode of Si at $\approx$520 cm-1. For sample #3, however, the Si line
was saturating the detector and thus it was acquired in a separate scan with
the incident power reduced by a factor 10. The Raman frequency and narrow
linewidth of the Si phonon peak from the substrates confirm a single-crystal
structure 42, 43.

Figure S1: Selected micro-Raman spectra showing the Si optical phonon peak
from the investigated samples (MoS2/Si membrane, MoS2/Si wafer and a MoS2 bulk
island). The Raman traces were rescaled and vertically shifted for clarity.
The blue lines correspond to the experimental data, while the red-dotted ones
to the best fit to a Voigt peak on top of a quadratic background.

To test the uniformity of the MoS2 film coverage for samples #2 and #3, we
performed a 5x5 raster scan, i.e., a series of consecutive Raman spectra taken
along a regular grid.

In Fig. S2, we show the 5x5 raster scan scheme employed for the micro-Raman
measurements. Panel (a) shows the coordinates of the employed grid (in µm),
superimposed to sample #2 in panel (b). An equivalent pattern was used to
characterize samples #1, #3 and a blank Si wafer. The spectra were fitted
using a series of Voigt peaks, one for each Raman resonance on top of a
quadratic background. The point by point results for the peak frequencies of
the two MoS2 Raman-active modes $E^{1}_{2g}$ and $A_{1g}$ and their separation
are shown in Fig. S3. In both sample #2 and #3, we observe an overall uniform
coverage with the values matching those of a monolayer film 1.

A picture of the multilayer, bulk-like, region found on the Si frame of sample
#2 is reported in Fig. S4. On such island we acquired the ‘bulk island’ traces
from Fig. 1(b) and S1.

The average frequency ($f$) and FWHM ($w$) of the MoS2 and Si peaks for the
studied samples are summarized in Tables 1, 2, together with the spectral
separation of the MoS2 peaks ($\Delta$).

Figure S2: (a) Scheme for the raster scan pattern to test the uniformity of
the MoS2 on the silicon substrate. The axes are in micrometers. (b) Optical-
microscope picture of sample #2 after changing the tonality and saturation
levels to emphasize the macroscopic morphology of the film. The light blue
section corresponds to the frame, while the pink regions to the membrane. The
membrane border is indicated by the green arrow. Figure S3: The left vertical
scale refers to the peak positions for the $E^{1}_{2g}$ and $A_{1g}$ MoS2
modes acquired at the positions of the raster scans (Fig. S2) through micro-
Raman measurements, while the right vertical scale refers to the spectral
separation between the two peaks; the linewidth of the Raman peaks was about 4
cm-1. (a) Sample #2 (b) Sample # 3. Figure S4: Multilayer, showing bulk-like
response, MoS2 region found on the Si frame of sample #2.

Sample $f_{E^{1}_{2g}}$ $w_{E^{1}_{2g}}$ $f_{A_{1g}}$ $w_{A_{1g}}$
$\Delta_{A_{1g}-E^{1}_{2g}}$ $f_{Si}$ $w_{Si}$ MoS2/Si m. $384.6\pm 0.1$
$4.02\pm{}0.08$ $404.2\pm 0.2$ $6.0\pm{}0.1$ $19.7\pm{}0.3$ $519.97\pm{}0.08$
$4.61\pm{}0.03$ MoS2/Si i. $381.83\pm{}0.04$ $3.83\pm{}0.07$ $407.03\pm{}0.02$
$4.57\pm{}0.04$ $25.20\pm{}0.05$ $519.29\pm{}0.01$ $4.94\pm{}0.02$ MoS2/Si w.
$385.4\pm{}0.1$ $4.26\pm{}0.04$ $404.4\pm{}0.2$ $6.4\pm{}0.2$ $18.9\pm{}0.4$
$520.91\pm{}0.04$ $4.55\pm{}0.05$

Table 1: Micro-Raman characterization results for regions with MoS2 film. All
the numerical results are expressed in cm-1. The abbreviation ‘m.’, ‘i.’ and
‘w.’ stand for ‘membrane’, ‘island’ and ‘wafer’ respectively.

Sample $f_{Si}$ $w_{Si}$ Si membrane $520.5\pm{}0.1$ $4.32\pm{}0.04$ Si wafer
$520.98\pm{}0.06$ $4.63\pm{}0.04$

Table 2: Micro-Raman characterization results for the filmless substrates. All
the numerical results are expressed in cm-1.

##  5 EUV/ Soft x-ray beam attenuation length

The EUV / Soft x-ray wavelength dependence of the attenuation length for
different materials is reported in Fig. S5. As shown by the logarithmic plot,
Si is the most transparent one under the wavelengths used in our transient
grating FEL experiments (13.3-39.9 nm). At $\lambda_{pu}$=39.9 nm, its
attenuation length is about one order of magnitude larger than the others,
which indicates that the pump beams travel through the whole Si membrane used
for the MoS2/Si membrane sample ($\approx$237 nm) without severe energy
reduction. A different case is, e.g., sapphire for which the attenuation
length is $\approx$13 nm, therefore meaning that a transient grating generated
by 39.9 nm beams will be localized close to the surface and leading to a much
higher energy density under the same excitation conditions.

Figure S5: EUV / Soft x-ray attenuation lengths for different materials. The
curves were calculated using the scattering factor data from 41.

##  6 Additional TG results

###  6.1 MoS2/Si wafer TG at different points

We report in Fig. S6 the TG response collected on the MoS2/Si wafer at
different positions on the sample. While we observed some variability in the
amplitude and decay time of the non-oscillating signal in the recorded window,
possibly due to the local morphology of both the substrate and the film, the
double peak structure and the general characteristics of the response are
maintained.

Figure S6: TG signal from the MoS2/Si wafer sample taken at different
positions from the dataset of Fig. 4(a) at $\approx$0.3 mJ/cm2 fluence.

###  6.2 Silicon frame

As a comparison, we show in Fig. S7 the TG signal obtained from the 200-µm-
thick supporting frame of the silicon membrane. While its bulk is still made
of silicon, the surface is covered by a thin oxide layer for surface
protection purposes.

Compared to the blank membrane response (Fig. 2(a)), the oscillatory response
displays a slower decay.

Figure S7: Transient grating signal acquired on the Si frame of the blank Si
membrane sample using a $83.6$ nm grating period acquired under 0.2 mJ/cm2
incident fluence respectively. The inset show the discrete Fourier transform
of the TG signal.

###  6.3 WS2/Sapphire

Fig. S8 shows in the EUV TG signal of a different heterostructure, a CVD-grown
monolayer WS2 on a $\approx$0.5-mm c-cut sapphire substrate (commercially
available from 2D Semiconductors), collected at different grating periods,
obtained by changing $\lambda_{pu}$ from (a) 39.9 nm ($L_{TG}$=83.6 nm) to (b)
26.6 nm ($L_{TG}$=55.8 nm) or (c) 13.3 nm ($L_{TG}$=27.9 nm). Differently from
the MoS2/Si samples examined in the main text, here we notice that a single
oscillatory term in Eq. 1 is enough to describe the TG response. In panel (d)
we show the obtained frequencies as a function of the transient grating
wavevector. Although collected at different fluences, we assume the acoustic
wave propagation velocities to be almost independent from it, similarly to the
MoS2/Si wafer data in Fig. 4(a). Through a linear fit, fixing the intercept to
0, we obtain an estimate propagation speed v=$(5.398\pm 0.009)$ km/s. This is
in good agreement with the Rayleigh mode of sapphire, reported to have a
velocity $\approx$5.4 km/s 44. Correspondingly, we have for the non-
oscillating background decay constant $\tau_{th}$ the following fit results:
(a) $(38\pm 2)$ ps (b) $(35\pm 2)$ ps (c) $(11\pm 3)$ ps. The last value is,
however, affected by the poorer TG signal signal-to-noise ratio connected to
the 13.3 nm pump - 13.3 probe signal, since this configuration is limited by
the impossibility to freely vary the relative fluence for pump and probe.
While in the second part of the explored time delay range (especially visible
in panels (a) and (b)), we observe a second harmonic modulation, connected to
the square in Eq. 2, the main frequency dominates elsewhere with a possible
contribution of the longitudinal-like mode the first tens of picoseconds,
similarly to the samples studied in the main text. The longitudinal speed in
sapphire is expected to be $\approx$11 km/s 45, corresponding to frequencies
of (a) $\approx$132 GHz (b) $\approx$198 GHz (c) $\approx$395 GHz. The reason
why the Rayleigh mode for this sample dominates the response even in the
presence of the WS2 film is likely connected to the much shallower penetration
of the pump beams in sapphire (Fig. S5) compared to Si, leading to a
significantly higher energy density near the surface, i.e. where the surface
acoustic waves such as the Rayleigh mode are localized. Moreover, as we saw in
Fig. 4(a) for the MoS2 / Si wafer sample, higher excitation fluences could
favor the emergence of the Rayleigh mode in the TG signal in WS2/Sapphire due
to larger initial displacements.

Figure S8: TG signal for WS2/Sapphire using different transient grating
periods. (a) 84 nm under 0.1 mJ/cm2 (b) 56 nm under 0.7 mJ/cm2 (c) 28 nm under
$<$0.1 mJ/cm2 (exact estimation not accurately available). (d) Frequency of
the Rayleigh mode as a function of the transient grating wavevector $k_{TG}$.
The experimental data is represented by blue traces, the fit curves by dotted
red lines.

##  7 Acoustic waves schemes

In Figs. S9 and S10 we report simulations of the propagation of the
longitudinal and Rayleigh mode on a semi-infinite bulk Si substrate at
different time delays within an oscillation period. The plots were drawn using
a regular 75 x 75 rectangular point grid. The (relative) displacements were
calculated based on the derivation reported in Royer and Valier-Brasier using
the method of potentials in the isotopic approximation (Chapter 1 and 3)19.
The displacements $\vec{u}$ were considered at different positions $x$
parallel to the surface and depths with respect to the surface ($y=0$,
positive values for larger depths in Si). For the longitudinal mode
propagating parallel to the surface, we used a simple sinusoidal form

$\begin{cases}u_{x}(x,y)=U\mathrm{cos}(k_{TG}x-\omega t)\\\
u_{y}(x,y)=0\end{cases}$ (2)

where $U$ is the amplitude of the mode, $k_{TG}$ the wavevector and $\omega$
is the angular frequency. In order to make the shifts visible, we set the
amplitude to an arbitrary value. In the plot, the color scale is normalized by
the largest magnitude among the calculated displacements.

Regarding the Rayleigh wave, the displacements are

$\begin{cases}Re[u_{x}(x,y)]=U_{1}(y)\mathrm{sin}(k_{TG}x-\omega t)\\\
Re[u_{y}(x,y)]=U_{2}(y)\mathrm{cos}(k_{TG}x-\omega t)\\\ \end{cases}$ (3)

where $\omega=k_{TG}v_{R}$ with $v_{R}$ Rayleigh velocity and the amplitudes

$\begin{cases}U_{1}(y)=a_{L}\left[-k_{TG}e^{-\alpha_{L}y}+\sqrt{\alpha_{L}\alpha_{T}}e^{-\alpha_{T}y}\right]\\\
U_{2}(y)=a_{L}\sqrt{\frac{\alpha_{L}}{\alpha_{T}}}\left[-\sqrt{\alpha_{L}\alpha_{T}}e^{-\alpha_{L}y}+k_{TG}e^{-\alpha_{T}y}\right]\\\
\end{cases}$ (4)

where $a_{L}$ is the longitudinal amplitude (connected to the transverse one
via $a_{T}=i\sqrt{\frac{\alpha_{L}}{\alpha_{T}}}a_{L}$),
$\alpha_{i}=\sqrt{k_{TG}^{2}-k_{i}^{2}}$ is a constant with
$k_{i}=\omega/v_{i}$ connected to the velocity $v_{i}$ with $i=l,t$
differentiating longitudinal and transverse parameters respectively. The
longitudinal, transverse velocities and Rayleigh values were taken from
Farnell and Adler 21. To ease visibility, the amplitude $a_{L}$ was set to an
arbitrary value.

Figure S9: Schemes of the displacements connected to a longitudinal wave in Si
at four different time instants, expressed as $\phi_{t}=\omega t$. Figure S10:
Schemes of the displacements connected to a Rayleigh wave in Si at four
different time instants, expressed as $\phi_{t}=\omega t$.

## References

  * Lee et al. 2010 Lee, C.; Yan, H.; Brus, L. E.; Heinz, T. F.; Hone, J.; Ryu, S. Anomalous Lattice Vibrations of Single- and Few-Layer MoS2. _ACS Nano_ 2010, _4_ , 2695–2700
  * Wang et al. 2012 Wang, Q. H.; Kalantar-Zadeh, K.; Kis, A.; Coleman, J. N.; Strano, M. S. Electronics and optoelectronics of two-dimensional transition metal dichalcogenides. _Nature Nanotechnology_ 2012, _7_ , 699–712
  * Golovynskyi et al. 2020 Golovynskyi, S.; Irfan, I.; Bosi, M.; Seravalli, L.; Datsenko, O. I.; Golovynska, I.; Li, B.; Lin, D.; Qu, J. Exciton and trion in few-layer MoS2: Thickness- and temperature-dependent photoluminescence. _Applied Surface Science_ 2020, _515_ , 146033
  * Ellis et al. 2011 Ellis, J. K.; Lucero, M. J.; Scuseria, G. E. The indirect to direct band gap transition in multilayered MoS2 as predicted by screened hybrid density functional theory. _Applied Physics Letters_ 2011, _99_ , 261908
  * Gabourie et al. 2022 Gabourie, A. J.; Köroǧlu, c. C.; Pop, E. Substrate-dependence of monolayer MoS2 thermal conductivity and thermal boundary conductance. _Journal of Applied Physics_ 2022, _131_ , 195103
  * Li et al. 2020 Li, G.; Chen, Z.; Li, Y.; Zhang, D.; Yang, W.; Liu, Y.; Cao, L. Engineering Substrate Interaction To Improve Hydrogen Evolution Catalysis of Monolayer MoS2 Films beyond Pt. _ACS Nano_ 2020, _14_ , 1707–1714
  * Agmon et al. 2022 Agmon, L.; Almog, R.; Gaspar, D.; Voscoboynik, G.; Choudhary, M.; Jopp, J.; Klausner, Z.; Ya’akobovitz, A.; Berkovich, R. Nanoscale contact mechanics of the interactions at monolayer MoS2 interfaces with Au and Si. _Tribology International_ 2022, _174_ , 107734
  * Kuppadakkath et al. 2022 Kuppadakkath, A. et al. Direct growth of monolayer MoS2 on nanostructured silicon waveguides. _Nanophotonics_ 2022, _11_ , 4397–4408
  * Hartmann 1985 Hartmann, C. S. Systems Impact of Modern Rayleigh Wave Technology. Rayleigh-Wave Theory and Application. Berlin, Heidelberg, 1985; pp 238–253
  * Biryukov et al. 1995 Biryukov, S. V.; Gulyaev, Y. V.; Krylov, V. V.; Plessky, V. P. _Surface Acoustic Waves in Inhomogeneous Media_ ; Springer-Verlag Berlin Heidelberg, 1995
  * Bergin et al. 2012 Bergin, S. M.; Chen, Y.-H.; Rathmell, A. R.; Charbonneau, P.; Li, Z.-Y.; Wiley, B. J. The effect of nanowire length and diameter on the properties of transparent, conducting nanowire films. _Nanoscale_ 2012, _4_ , 1996–2004
  * Leong et al. 2015 Leong, W. S.; Luo, X.; Li, Y.; Khoo, K. H.; Quek, S. Y.; Thong, J. T. L. Low Resistance Metal Contacts to MoS2 Devices with Nickel-Etched-Graphene Electrodes. _ACS Nano_ 2015, _9_ , 869–877
  * Ganatra and Zhang 2014 Ganatra, R.; Zhang, Q. Few-Layer MoS2: A Promising Layered Semiconductor. _ACS Nano_ 2014, _8_ , 4074–4099
  * Nelson et al. 1982 Nelson, K. A.; Miller, R. J. D.; Lutz, D. R.; Fayer, M. D. Optical generation of tunable ultrasonic waves. _Journal of Applied Physics_ 1982, _53_ , 1144–1149
  * Bencivenga et al. 2019 Bencivenga, F. et al. Nanoscale transient gratings excited and probed by extreme ultraviolet femtosecond pulses. _Science Advances_ 2019, _5, 7, eaaw5805_
  * Liu et al. 2020 Liu, F.; Wu, W.; Bai, Y.; Chae, S. H.; Li, Q.; Wang, J.; Hone, J.; Zhu, X.-Y. Disassembling 2D van der Waals crystals into macroscopic monolayers and reassembling into artificial lattices. _Science_ 2020, _367_ , 903–906
  * Soranzio 2024 Soranzio, D. Supplementary information. 2024
  * Auld 1973 Auld, B. A. _Acoustic fields and waves in solids. Vol. 2_ ; John Wiley I& Sons, New York, 1973
  * Royer and Valier-Brasier 2022 Royer, D.; Valier-Brasier, T. _Elastic Waves in Solids 1: Propagation_ ; ISTE, London and John Wiley & Sons, New York, 2022
  * Milloch 2020 Milloch, A. Nanoscale thermoelasticity in mechanically confined systems. M.Sc. thesis, Università degli studi di Trieste, 2020
  * Farnell and Adler 1972 Farnell, G.; Adler, E. _Elastic Wave Propagation in Thin Layers_ ; Physical Acoustics; Academic Press, 1972; Vol. 9; pp 35–127
  * Janušonis et al. 2016 Janušonis, J.; Jansma, T.; Chang, C. L.; Liu, Q.; Gatilova, A.; Lomonosov, A. M.; Shalagatskyi, V.; Pezeril, T.; Temnov, V. V.; Tobey, R. I. Transient Grating Spectroscopy in Magnetic Thin Films: Simultaneous Detection of Elastic and Magnetic Dynamics. _Scientific Reports_ 2016, _6_ , 29143
  * Maznev et al. 2018 Maznev, A. A. et al. Generation of coherent phonons by coherent extreme ultraviolet radiation in a transient grating experiment. _Applied Physics Letters_ 2018, _113_ , 221905
  * Brioschi et al. 2023 Brioschi, M.; Carrara, P.; Polewczyk, V.; Dagur, D.; Vinai, G.; Parisse, P.; Zilio, S. D.; Panaccione, G.; Rossi, G.; Cucini, R. Multidetection scheme for transient-grating-based spectroscopy. _Opt. Lett._ 2023, _48_ , 167–170
  * Foglia et al. 2023 Foglia, L. et al. Extreme ultraviolet transient gratings: A tool for nanoscale photoacoustics. _Photoacoustics_ 2023, _29_ , 100453
  * Bencivenga et al. 2023 Bencivenga, F.; Capotondi, F.; Foglia, L.; Mincigrucci, R.; Masciovecchio, C. Extreme ultraviolet transient gratings. _Advances in Physics: X_ 2023, _8_ , 2220363
  * Janušonis et al. 2016 Janušonis, J.; Chang, C. L.; Jansma, T.; Gatilova, A.; Vlasov, V. S.; Lomonosov, A. M.; Temnov, V. V.; Tobey, R. I. Ultrafast magnetoelastic probing of surface acoustic transients. _Phys. Rev. B_ 2016, _94_ , 024415
  * Xu et al. 2004 Xu, B.; Shen, Z.; Ni, X.; Lu, J. Numerical simulation of laser-generated ultrasound by the finite element method. _Journal of Applied Physics_ 2004, _95_ , 2116–2122
  * Veres et al. 2013 Veres, I. A.; Berer, T.; Burgholzer, P. Numerical modeling of thermoelastic generation of ultrasound by laser irradiation in the coupled thermoelasticity. _Ultrasonics_ 2013, _53_ , 141–149
  * Liu et al. 2021 Liu, Z.; Lin, B.; Liang, X.; Du, A. Numerical simulation of laser-generated Rayleigh wave pulses propagation in the machined surface with residual stress using finite-difference method. _Optik_ 2021, _248_ , 168072
  * Wang et al. 2021 Wang, J.; Wu, S.; Xie, H.; Xiong, L. Theoretical study on thermal properties of molybdenum disulfide/silicon heterostructures. _Computational Materials Science_ 2021, _200_ , 110835
  * Liu et al. 2023 Liu, W.; Huang, X.; Yue, Y. Tuning thermal transport across monolayer MoS2/Si heterostructure via substrate nanogrooving. _International Journal of Heat and Mass Transfer_ 2023, _201_ , 123673
  * Johnson et al. 2013 Johnson, J. A.; Maznev, A. A.; Cuffe, J.; Eliason, J. K.; Minnich, A. J.; Kehoe, T.; Torres, C. M. S.; Chen, G.; Nelson, K. A. Direct Measurement of Room-Temperature Nondiffusive Thermal Transport Over Micron Distances in a Silicon Membrane. _Phys. Rev. Lett._ 2013, _110_ , 025901
  * Maznev et al. 2011 Maznev, A. A.; Johnson, J. A.; Nelson, K. A. Onset of nondiffusive phonon transport in transient thermal grating decay. _Phys. Rev. B_ 2011, _84_ , 195206
  * Zhao and Ouyang 2019 Zhao, Y.; Ouyang, G. Thickness-dependent photoelectric properties of MoS2/Si heterostructure solar cells. _Scientific Reports_ 2019, _9_ , 17381
  * Vialla and Fatti 2020 Vialla, F.; Fatti, N. D. Time-Domain Investigations of Coherent Phonons in van der Waals Thin Films. _Nanomaterials_ 2020, _10, 12, 2543_
  * 37 Norcada: a MEMS technology company. https://www.norcada.com/, Accessed: 2023-11-07
  * Mincigrucci et al. 2018 Mincigrucci, R. et al. Advances in instrumentation for FEL-based four-wave-mixing experiments. _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 2018, _907_ , 132–148, Advances in Instrumentation and Experimental Methods (Special Issue in Honour of Kai Siegbahn)
  * Bencivenga et al. 2015 Bencivenga, F. et al. Four-wave mixing experiments with extreme ultraviolet transient gratings. _Nature_ 2015, _520_ , 205–208
  * Allaria et al. 2013 Allaria, E. et al. Two-stage seeded soft-X-ray free-electron laser. _Nature Photonics_ 2013, _7_ , 913–918
  * Henke et al. 1993 Henke, B.; Gullikson, E.; Davis, J. X-Ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92. _Atomic Data and Nuclear Data Tables_ 1993, _54_ , 181–342
  * Lambertz et al. 1998 Lambertz, A.; Richards, B.; Sproul, A.; Puzzer, T.; Gross, M. Sputter Deposited and Solid Phase Crystallized Silicon Films for Solar Cells. Proceedings of the 2nd World Conference on Photovoltaic Solar Energy Conversion. 1998; p 1294
  * Yeh 2017 Yeh, W. Fabrication of single crystalline stripe in Si and Ge film on rolled flexible glass substrate by UV cw micro-chevron laser beam. Low-Dimensional Materials and Devices 2017. 2017; pp 31 – 37
  * Pedrós et al. 2004 Pedrós, J.; Calle, F.; Grajal, J.; Jiménez Riobóo, R.; Prieto, C.; Pau, J.; Pereiro, J.; Hermann, M.; Eickhoff, M.; Bougrioua, Z. Anisotropic propagation of surface acoustic waves on nitride layers. _Superlattices and Microstructures_ 2004, _36_ , 815–823, European Materials Research Society 2004, Symposium L. InN, GaN, AlN and Related Materials, their Heterostructures and Devices
  * Winey et al. 2001 Winey, J. M.; Gupta, Y. M.; Hare, D. E. r-axis sound speed and elastic properties of sapphire single crystals. _Journal of Applied Physics_ 2001, _90_ , 3109–3111