Time-resolved detection of propagating Lamb waves in thin silicon
membranes with frequencies up to 197 GHz
Martin Grossmann,
1,a)
Oliver Ristow,
1
Mike Hettich,
1
Chuan He,
1
Reimar Waitz,
1
Elke Scheer,
1
Vitalyi Gusev,
2
Thomas Dekorsy,
1
and Martin Schubert
1
1
Department of Physics, University of Konstanz, 78457 Konstanz, Germany
2
LUNAM Universit
es, CNRS, Universit
e du Maine, LAUM UMR-CNRS 6613, Av. O. Messiaen,
72085 Le Mans, France
(Received 6 February 2015; accepted 16 April 2015; published online 27 April 2015)
Guided acoustic waves are generated in nanopatterned silicon membranes with aluminum gratings
by optical excitation with a femtosecond laser. The spatial modulation of the photoacoustic
excitation leads to Lamb waves with wavelengths determined by the grating period. The excited
Lamb waves are optically detected for different grating periods and at distances up to several lm
between pump and probe spot. The measured frequencies are compared to the theoretical
dispersion relation for Lamb waves in thin silicon membranes. Compared to surface acoustic
waves in bulk silicon twice higher frequencies for Lamb waves (197 GHz with a 100 nm grating)
are generated in a membrane at equal grating periods.
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2015 AIP Publishing LLC.
[http://dx.doi.org/10.1063/1.4919132]
Lamb waves are acoustic waves guided between the
mechanically free opposite surfaces of an elastic plate.
Various methods for the generation and detection of Lamb
waves by lasers have been developed for applications in
noncontact nondestructive testing, in evaluation of thin films,
and in free-standing plates.
1–5
These investigations include
the all-optical monitoring of zero-group-velocity Lamb
modes as demonstrated recently by Prada et al.
4,5
Continuous progress in the development of methods for the
fabrication of membranes with a thickness in the nm range
for a variety of applications
6
requires the evaluation of their
parameters and quality, e.g., their dispersion relation and
damping, the development of methods for both the genera-
tion and detection of acoustic waves at GHz frequencies.
Standing waves in free-standing films of lm thickness
have been first detected by Brillouin scattering.
7
Recently,
femtosecond lasers have been successfully applied to gener-
ate and detect longitudinal acoustic modes in free-standing
silicon (Si) membranes with frequencies up to 500 GHz
corresponding to 20 nm wavelengths.
8–10
As the dispersion
relations of propagating Lamb waves are emerging from
standing wave resonances,
11
these experiments indicate that
propagating Lamb waves in membranes at similarly high
frequencies could be monitored all-optically as well, if the
conditions for their effective generation and detection are
achieved. Recently, Rayleigh waves with wavelengths down
to 45 nm (Refs. 12 and 13) and frequencies up to 90 GHz
(Ref. 12) have been generated and detected based on the dep-
osition of a metallic grating of nanometer periodicity on the
substrate surface. Rayleigh waves are surface acoustic waves
on a semi-infinite substrate.
14
Here, we combine the femto-
second pump-probe technique applied earlier to the monitor-
ing of standing wave resonances
8–10
and of Rayleigh
waves
12
with nanostructuring of the surface
12,13,15
in order
to achieve the generation and detection of propagating Lamb
waves at frequencies up to 197 GHz and wavelengths down
to 50 nm in thin semiconductor membranes. Those high fre-
quencies can be generated because for the same wave vectors
the dispersion relations of Lamb waves contain modes of
much higher frequencies than those of Rayleigh waves. The
highest frequencies of the detected coherent Lamb modes
are nearly three times higher than the highest frequencies of
thermal Lamb modes detected by classical Brillouin
scattering.
16
Different samples with aluminum (Al) gratings from
100 nm to 500 nm period are prepared on crystalline Si mem-
branes (326 nm thickness). The grating stripes are oriented
perpendicular to the [110]-direction on the (001)-surface of
Si with a filling factor of 50%. The thickness of the Al is
17 nm and the stripes are partly embedded in the membrane
(see Fig. 1). A part of the membrane is covered by a uniform
Al film for reference measurements. Details regarding the
fabrication of the membranes and the gratings are discussed
in the previous publications.
12,17
The method and setup used are similar to the one previ-
ously employed for the generation and detection of Rayleigh
waves.
12
We excite and detect the Lamb waves with an
FIG. 1. Pump-probe scheme of the measurement on a thin Si membrane
with an Al grating. Pump and probe pulses are laterally displaced.
a)
martin.grossmann@uni-konstanz.de
0003-6951/2015/106(17)/171904/4/$30.00
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2015 AIP Publishing LLC106, 171904-1
APPLIED PHYSICS LETTERS 106, 171904 (2015)
Konstanzer Online-Publikations-System (KOPS)
URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-288114
Erschienen in: Applied Physics Letters ; 106 (2015), 17. - 171904
optical pump-probe technique based on high-speed asyn-
chronous optical sampling (ASOPS) with femtosecond lasers
at GHz repetition rate.
18
The measurements are performed in
a confocal microscope configuration, in which the laser spots
can be focussed below 2 lm full width at half maximum
(FWHM) on the sample surface. The pump (probe) polariza-
tion is perpendicular (parallel) to the grating stripes. For
measurements of the propagation of the Lamb waves, pump
and probe spots can be spatially displaced relative to each
other in the focal plane. The wavelengths for the pump and
probe are 790 nm with a power of 20 mW and 820 nm with a
power of 3.5 mW, respectively. For the single
stripe measurement, one and a half and four times larger
intensities were used in the pump and the probe laser beams,
respectively.
Fig. 2 shows three different temporal transients for a
grating wit h 100 nm period, corresponding to three different
distances between pump and probe spots. The top graph
shows the measurement with spatially overlapping pump and
probe spots. A strong change in reflectivity can be seen in
the beginning of the transient, marking the time at which the
pump pulse excites electron-hole pairs in the Si membrane.
Superimposed on the decay of the electronic contribution are
oscillations, which are shown in the inset. The background is
subtracted in order to improve the visibility of the oscillatory
components. These are coherently excited acoustic phonons
representing acoustic eigenmodes of the membrane. The
middle and bottom graphs show two measurements with dis-
tances of 6.8 lm and 7.8 lm between the pump and probe
spots. The oscillation maxima (the center of the wavepackets
indicated by the black arrows) are centered around 300 ps
for the shorter and 800 ps for the longer distances. The
temporal difference between the two maxima reflects the
additional propagation length, which the acoustic wave
packet needs to travel in the membrane from the pump to the
probe spot. The 1 lm larger distance results in 500 ps delay,
which corresponds to a group velocity of around 2000 m/s.
The acoustic wave packet would need for the 6.8 lm dis-
tance, 3.4 ns. We can therefore conclude that due to the high
repetition rate of the lasers (800 MHz), the measured signal
stems from a preceding pump pulse at a multiple of the
inverse repetition rate (1.25 ns). The propagating Lamb
modes are discussed in more detail below.
In Figs. 3(a)–3(c), the Fourier spectra for the 100 nm
grating corresponding to the time-domain data in Fi g. 2 are
shown. The spectrum where both pump and probe are spa-
tially overlapping (a) shows a dense frequency distribution.
The two spectra ((b) and (c)), taken at a distance of 6.8 and
7.8 lm between pump and probe, show a strong change of
the amplitude ratios of different spectral components and a
less dense mode spectrum; a total of sixteen modes can be
clearly resolved. As previously observed for Rayleigh waves,
we assume that the grating structure excites wavevectors
corresponding to the inverse grating period.
12
Additionally,
the Fourier analysis reveals that odd spatial harmonics are
excited due to the fact that the grating is not sinu soidal but
rectangular, while even spatial harmonics can be excited
because of defects in the grating periodicity.
The observed frequencies are plotted in Fig. 4 for two
different wave vectors, i.e., their corresponding grating pe-
riod: the basic grating period and the first even harmonic
(black diamonds). Further, we plot the Lamb dispersion rela-
tion for a thin Si plate along the [110]-direction.
19
By look-
ing at the basic wave vector, most of the measured
frequencies can be clearly assigned to a mode given by the
Lamb dispersion. Slight discrepancies can be explained by
the measurement system: the sample is excited every 1.25 ns
with 800 MHz, which limits the frequency resolution to
around 1 GHz and also limits the determination of the wave-
vector due to the finite length of the wave packet excited
with the grating; the grating is illuminated with a pump spot
of less than 2 lm FWHM. When the grating period gets
smaller, the error due to the finite laser spot decreases. As
well, the fabrication process gives both a variation in grating
period and membrane thickness (65 nm), leading to an
increased error for smaller grating periods. Both
FIG. 2. Time-resolved modulation of the reflected probe beam DR/R
0
of the
326 nm Si membrane with Al grating (17 nm thickness and 100 nm period)
on top. The transient at the top shows the modulation when pump and probe
spots are overlapped on the grating. The transient in the middle (bottom)
shows the modulation when pump and probe are separated by 6.8 l m
(7.8 lm). The arrows indicate the position of the center of the individual
wavepackets. Note a change in the amplitude of DR/R
0
by three orders of
magnitude between top and middle/bottom graphs.
FIG. 3. (a)–(c) Fourier spectra of the different time transients in Fig. 2. (d)
The damping distance over frequency is plotted for the 100 nm and the
500 nm grating and the expected frequencies given by the Lamb dispersion
relation are shown by vertical lines (for symmetric modes in red and for
antisymmetric in black).
171904-2 Grossmann et al. Appl. Phys. Lett. 106, 171904 (2015)
contributions lead to a nearly homogeneous error of 5%–6%
for the wavevector of every grating period.
There are some frequencies, however, that cannot be
attributed to a frequency given by the dispersion relation.
Here, the higher harmonics are important. We checked, for
example, that the experimental frequency of 196.8 GHz is
closer to the theoretical frequency of 196.3 GHz of the 15th
asymmetric Lamb mode with the second harmonic of the
grating period than to the frequency of 192.1 GHz of the
19th symmetric Lamb mode with the basic grating period.
The second harmonic therefore means that this mode has a
wavelength of 50 nm.
In the spectra with the spatially separated pump and probe,
the 58.4 GHz mode has the largest amplitude. It corresponds to
a Lamb wave of 100 nm wavelength with a phase velocity of
about 5840 m/s, which is close to the transversal speed of
sound in bulk Si. We attribute the large amplitude to the fact
that in the dispersion relation, two Lamb modes are close to-
gether in the vicinity of this frequency and wavevector.
There are two modes which cannot be explained by any
of the harmonics: at 24.9 and 44.1 GHz. Due to the spatial
width of the pump laser spot we also excite modes with a
small wavevector, quite smaller than for the grating period
itself. As can be seen in Fig. 4, two modes can be fo und at
small wave vectors that match the measured frequencies
very well. A priori we cannot distinguish any of the meas-
ured frequencies by their excitation processes (finite spot
size, grating period, and higher harmonic). However, as
explained above, for some of the individual frequencies we
are sure that they correspond to higher harmonics of the gra-
ting period.
Figure 3(d) shows the damping for the individual modes.
The damping distance, i.e., the distance over which the am-
plitude reaches 1/e, was evaluated by fitting the FFT ampli-
tudes with an exponential decay over distance. For the lower
frequencies, damping distances around 2 lm are found,
while for the higher frequencies lower damping distances
around 1 lm are found.
We perform the same experiments on samples with gra-
ting periods of 200 to 500 nm in 100 nm steps. Up to 21
modes are visible. All observed frequencies are plotted in
Fig. 4 with respect to their grating period including the first
three harmonics. Because it is not possible to distinguish,
which mode belongs to which wavevector, we did not assign
any mode to an individual wavevector. As for the 100 nm
grating case, most of the measured frequencies can be clearly
assigned to a frequency given by the Lamb dispersion.
The damping distance for the 500 nm grating is shown in
Fig. 3(d) together with the 100 nm case. For both structures,
a similar frequency dependent damping behavior can
be seen.
The measured Lamb modes are excited due to two dif-
ferent physical mechanisms: one is the thermo-elastically
generated stress in the Al grating, leading to a local negative
strain, and the other one is due to the deformation potential
in Si, leading to a posi tive strain. Both effects need to be
taken into consideration. The periodic expansion of the Al
metal stripes on top of the grating as well as the compression
of the underlying membrane, which is being masked for opti-
cal excitation by the grating on top,
20
also lead to a periodic
modulation. To calculate both contributions is beyond the
scope of this article, as not only the strain is difficult to
calculate due to carrier diffusion
21,22
but also the optical field
that excites the structure.
23,24
We attribute the absence of some expected modes to the
mutual compensation of two different contributions to the
detection process: Two physical mechanisms play a role in
the detection of the Lamb mode. These are the photoelastic
effect in Al and Si and the interferometric effect in the Si
membrane. The interferometric effect is caused by surface
corrugation due to the Lamb waves, i.e., the relative motion
of the opposite surfaces, which causes modification of the
membrane thickness. Quite recently, it has been demon-
strated
25
that the optical detection process of confined
eigenmodes can be either enhanced or reduced to zero fol-
lowing a selection rule. In particular, it has been shown that
detection configurations exist where the coherent acoustic
phonons and probe light sensitivity functions are orthogonal,
leading to cancelation of the photoelas tic detection
mechanism.
We investigate the detection process more closely with
a different sample structure. The excitation was performed
on two (nominal) 100 nm gratings on the same wafers (A1
and A2). We first compared these nominally identical
gratings by detecting the spectra of their vibrations with the
pump and probe beams overlapping on the gratings (see top
panel in Fig. 5). This comparison demonstrates that the main
spectral characteristics are well reproduced especially at fre-
quencies of propagating Lamb modes (above 50 GHz). Then,
we detected the Lamb spectra in the case where the pump
and probe beams are spatially separated. In sample A1, the
Lamb waves were propagating all their path of 6.8 lm along
the grating, while in the sample A2 they propagated 5 lm
along the grating and 10 lm along the free surface of the Si
membrane between the edge of the grating and a single
detecting stripe (width: 1.5 lm). The comparison of the
detected spectra in the bottom panel of Fig. 5 indicates that
the metallic grating enhances the optical detection especially
of high frequency Lamb modes.
FIG. 4. Dispersion relation of Lamb modes in a thin Si membrane in the
[110]-direction. Black lines mark the symmetric modes and red lines mark
the asymmetric ones. The data points correspond to the first, second, and
third harmonic of the grating period (left to right) and the corresponding
measured frequencies (see Fig. 3).
171904-3 Grossmann et al. Appl. Phys. Lett. 106, 171904 (2015)
One also has to keep in mind that the Al grating may not
only allow to excite ultra-high frequency modes but also
affect the eigenmodes of the membrane. Although the differ-
ence in longitudinal impedances of Al and Si is negligible,
the difference in shear impedances is significant and, thus,
the modification of the spectra of the Lamb modes in the
presence of the Al grating is to be expected. However, it has
been demonstrated earlier
12
that even for the Rayleigh waves
of 100 nm length localized within 100 nm from the surface
covered by a similar Al grating, the frequency shift does not
exceed 10%. The higher order Lamb modes, guided by the
complete thickness of the membrane (324 nm), that are over-
lapping with the fingers are more than three times weak er
than that. Thus, we would expect that the frequency modifi-
cation caused by the grating does not exceed 5%–6%. These
effects should be taken into account if a precise identification
of the Lamb modes is necessary.
We showed that thin Si membranes represent an excel-
lent system to generate guided waves in the hypersonic
frequency range. For the same wavelength, higher-order
Lamb modes are of higher frequency in comparison with
Rayleigh waves. This provides the opportunity to generate
higher frequencies by laser-excited structures with the same
spatial modulation. We were able to generate frequencies up
to 197 GHz with wavelengths down to 50 nm.
There is currently growing interest to the application of
the resonances of guided acoustic waves in nanocantilevers
for the evaluation of liquids,
26,27
to the interaction of
Rayleigh surface acoustic waves with other types of excita-
tions, and also with charge carriers in the vicinity of the solid
surface (see Refs. 28–30 and the references therein). Thus,
we can reasonably expect that Lamb waves of hundreds of
GHz frequencies and tens of nanometers wavelengths would
find applications not onl y in the testing and evaluation of
thin membranes but also for studying the fundamental prob-
lems of liquid rheology at GHz frequencies through experi-
ments with liquid-loaded membranes and for probing the
interaction of travelling acoustic waves with charge carriers
and other type of excitations in solids in nanometer-confined
geometries.
31
This work was supported by the DFG through the SFB
767 (Germany) and by the Ministry of Science, Research,
and Arts of Baden-W
€
urttemberg (Germany). The authors
also thank Matthias Hagner for providing support in the
nanostructure laboratory.
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FIG. 5. Fourier spectra of two measurements where the pump is on two dif-
ferent 100 nm gratings. The top measurement was done with pump and
probe overlapped for two different samples (A1 and A2) with nominally the
same properties. The bottom spectra are taken at a distance between pump
and probe spots with the pump spot on the grating. The probe spot is on the
grating in 6.8 lm distance (A1) or on a 1.5 lm wide single stripe deposited
in 10 lm distance parallel to the grating (A2) with 15 lm distance to the
pump spot.
171904-4 Grossmann et al. Appl. Phys. Lett. 106, 171904 (2015)
![FIG. 4. Dispersion relation of Lamb modes in a thin Si membrane in the [110]-direction. Black lines mark the symmetric modes and red lines mark the asymmetric ones. The data points correspond to the first, second, and third harmonic of the grating period (left to right) and the corresponding measured frequencies (see Fig. 3).](/figures/fig-4-dispersion-relation-of-lamb-modes-in-a-thin-si-291727zw.webp)
