Publisher’s Note: Mode shape and dispersion relation of bending waves in thin silicon membranes [Phys. Rev. B<b>85</b>, 035324 (2012)]

Waitz, Reimar; Nößner, Stephan; Hertkorn, Michael; Schecker, Olivier; Scheer, Elke

doi:10.1103/physrevb.86.039904

Cited by 5 publications

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“…A number of methods have evolved to visualize the vibrational mode shapes, such as scanning vibrometry, 16,17 holographic interferometry, 18 and stroboscopic interferometry. 19,20 Among those, stroboscopic interferometry is capable of visualizing 3D vibrational modes with a lateral resolution in the sub-micrometer range and a vertical resolution in the sub-nanometer range, as exemplified by measurements performed on a 340 nm thick Si membrane. 20 Carbon nanomembranes (CNMs) are a class of twodimensional materials that are fabricated by cross-linking self-assembled monolayers (SAMs) of aromatic precursor molecules via low-energy electron irradiation.…”

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“…In ambient conditions, the membrane oscillations would be strongly damped by the emission of sound waves and the inertia of the air surrounding the membrane would make the analysis of the vibration more complicated. 20 Therefore, the sample was located inside a vacuum cell. A pressure of about 4 mbar is achieved in our setup to suppress these complications.…”

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“…More details about excitation and detection of membrane waves have been described elsewhere. 20 For a NM with a thickness of d, the bending stiffness can be determined from the equation D ¼ Ed 3 =12ð1 À 2 Þ, with E being the Young's modulus and the Poisson's ratio. The bending stiffness of a single-layer NBPT-CNM is estimated to be 8:5 Â 10 À19 N m, which is even four times smaller than the value 3:6 Â 10 À18 N m of graphene.…”

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“…The data points of the dispersion relations were obtained from the measured data by searching for the highest amplitude Q mn for any givenk vector and the so called "maximum amplitude method" is explained in detail elsewhere. 20 Each data point denotes a vibrational mode with 24 mode shapes measured at different phases. As an example, the five points labeled by red circles in Fig.…”

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confidence: 99%

“…3(c)). In general, the scattering of data points in the dispersion relations can be due to uncertainties in the determination of k, to anisotropic stress, 20,33 as well as to inhomogeneity in the area density as a result of the fabrication processes (see dotted-like PMMA residue in Fig. 1(d)).…”

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Vibrational modes of ultrathin carbon nanomembrane mechanical resonators

Zhang

Waitz

Yang

et al. 2015

Applied Physics Letters

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We report measurements of vibrational mode shapes of mechanical resonators made from ultrathin carbon nanomembranes (CNMs) with a thickness of approximately 1 nm. CNMs are prepared from electron irradiation induced cross-linking of aromatic self-assembled monolayers and the variation of membrane thickness and/or density can be achieved by varying the precursor molecule. Singleand triple-layer freestanding CNMs were made by transferring them onto Si substrates with square/ rectangular orifices. The vibration of the membrane was actuated by applying a sinusoidal voltage to a piezoelectric disk on which the sample was glued. The vibrational mode shapes were visualized with an imaging Mirau interferometer using a stroboscopic light source. Several mode shapes of a square membrane can be readily identified and their dynamic behavior can be well described by linear response theory of a membrane with negligible bending rigidity. By applying Fourier transformations to the time-dependent surface profiles, the dispersion relation of the transverse membrane waves can be obtained and its linear behavior verifies the membrane model. By comparing the dispersion relation to an analytical model, the static stress of the membranes was determined and found to be caused by the fabrication process. V C 2015 AIP Publishing LLC.

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Vibrational modes of ultrathin carbon nanomembrane mechanical resonators

Zhang

Waitz

Yang

et al. 2015

Applied Physics Letters

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Imaging Off‐Resonance Nanomechanical Motion as Modal Superposition

et al. 2021

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Observation of resonance modes is the most straightforward way of studying mechanical oscillations because these modes have maximum response to stimuli. However, a deeper understanding of mechanical motion can be obtained by also looking at modal responses at frequencies in between resonances. Here, an imaging of the modal responses for a nanomechanical drum driven off resonance is presented. By using the frequency modal analysis, these shapes are described as a superposition of resonance modes. It is found that the spatial distribution of the oscillating component of the driving force, which is affected by both the shape of the actuating electrode and inherent device properties such as asymmetry and initial slack, greatly influences the modal weight or participation. This modal superposition analysis elucidates the dynamics of any nanomechanical system through modal weights. This aids in optimizing mode-specific designs for force sensing and integration with other systems.

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Non‐Uniform Strain Engineering of 2D Materials

Kovalchuk

Kirchhof

Bolotin

et al. 2022

Israel Journal of Chemistry

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2D materials are elastic substances that can sustain high strain. While the response of these materials to spatially uniform strain is well studied, the effects of spatially non‐uniform strain are understood much less. In this review, we examine the response of two different 2D materials, transition metal dichalcogenides and graphene, under non‐uniform strain. First, we analyze pseudo‐magnetic fields formed in graphene subjected to highly localized non‐uniform strain. Second, we discuss the effect of non‐uniform strain on excitons in non‐uniformly strained TMDC. We show that while transport or “funneling” of excitons is relatively inefficient, a different process, a strain‐related conversion of excitons to trions is dominant. Finally, we discuss the effects of uniform and non‐uniform strain in a graphene‐based phononic crystal. We find that uniform strain can be used to broadly tune the frequency of the phononic bandgap by more than 350 % and non‐uniform strain smears that bandgap.

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Publisher’s Note: Mode shape and dispersion relation of bending waves in thin silicon membranes [Phys. Rev. B85, 035324 (2012)]

Abstract: Publisher's Note: Mode shape and dispersion relation of bending waves in thin silicon membranes [Phys. Rev. B 85, 035324 (2012)]

Cited by 5 publications

References 0 publications

Vibrational modes of ultrathin carbon nanomembrane mechanical resonators

Vibrational modes of ultrathin carbon nanomembrane mechanical resonators

Imaging Off‐Resonance Nanomechanical Motion as Modal Superposition

Non‐Uniform Strain Engineering of 2D Materials

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