Realization and direct observation of five normal and parametric modes in silicon nanowire resonators by in situ transmission electron microscopy

Abstract

Mechanical resonators have wide applications in sensing bio-chemical substances, and provide an accurate method to measure the intrinsic elastic properties of oscillating materials. A high resonance order with high response frequency and a small resonator mass are critical for enhancing the sensitivity and precision. Here, we report on the realization and direct observation of high-order and high-frequency silicon nanowire (Si NW) resonators. By using an oscillating electric-field for inducing a mechanical resonance of singlecrystalline Si NWs inside a transmission electron microscope (TEM), we observed resonance up to the 5 th order, for both normal and parametric modes at $100 MHz frequencies. The precision of the resonant frequency was enhanced, as the deviation reduced from 3.14% at the 1 st order to 0.25% at the 5 th order, correlating with the increase of energy dissipation. The elastic modulus of Si NWs was measured to be $169 GPa in the [110] direction, and size scaling effects were found to be absent down to the $20 nm level.

Intrinsic and Defect-Related Elastic Moduli of Boron Nitride Nanotubes As Revealed by in Situ Transmission Electron Microscopy

“…Taking fourth order as an example, theoretically calculated nodes are at 0.358 L , 0.644 L , and 0.906 L , whereas experimental values are 0.368 L , 0.649 L , and 0.915 L , with an average difference of ∼1.5%. As also could be seen from additional results of BNNTs with high-order resonances (Figures S2 and S3), the precision of the measurement method is ∼1%, consistent with our previous reports …”

“…According to eq , the corresponding elastic moduli were calculated to be 922.4, 908.1, 922.4, and 989.9 GPa, at each resonance frequency. Figure b shows the first to third parametric resonances of the same BNNT at frequencies twice of the corresponding natural resonance, as predicted by the Mathieu equation Figure c shows the relationship between resonance frequencies and mode orders.…”

“…The elastic moduli were calculated from the resonance frequencies at high-order resonance modes with the precision ∼1%, as reported in our previous works. , According to the classical Euler–Bernoulli (E-B) equation for a cantilevered beam and considering the hollow cross-section of BNNT, the natural resonance frequency f i can be expressed as − Therefore, the elastic modulus E is given by where D o , D i , ρ, and L are the outer and inner tube diameters, mass density, and calibrated length of the nanotube (Figure S1). β i is an eigenvalue, where β 1 = 1.875, β 2 = 4.694, β 3 = 7.855, β 4 = 10.996, and the integer subscripts i = 1, 2, ... correspond to the first, second, and higher natural resonance orders .…”

“…In addition, resonance modes could be also excited at the frequencies of f ni = , which was described by the Mathieu equation. These resonance modes were denoted as parametric modes, and n is defined as the number of parametric order ( n = 1, 2, 3, ...) …”

Nanomass Sensing via Node Shift Tracing in Vibrations of Coupled Nanowires Enhanced by Fano Resonances

“…An alternative is to employ in situ visualization of NW vibrations in scanning electron , or transmission electron , microscopes (SEM and TEM, respectively). The resonance frequency shift is typically acquired from the amplitude measurements during the frequency sweep.…”