Surface stress effect on bending resonance of nanowires with different boundary conditions

He, Jin; Lilley, Carmen M.

doi:10.1063/1.3050108

Cited by 174 publications

(144 citation statements)

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“…The fundamental resonant frequency of a fixed-fixed gold nanowire and a cantilever one with (001) surface was calculated numerically by Park and Klein 24 based on a SCB model, which is compared to the theoretical prediction by the present model and the Young-Laplace (Y-L) one 35 as shown in Fig. 3.…”

Section: Resultsmentioning

confidence: 99%

“…For simplicity, we assume a [100] axially oriented nanowire with a symmetric lateral surface in the present model, which has an equal atom spacing in both bond directions, e.g., the (001) or (010) surface. 13,34,35 In such a case, the Lagrangian surface energy density can be written as 45…”

Section: A Variational Analysismentioning

confidence: 99%

“…In the following research, only the smallest positive solution of the two equations is numerically solved, which is referred as the angular resonant frequency for the first vibration modes (fundamental resonant frequency). 35,58 Meanwhile, according to the conventional structural dynamics, 59 the fundamental frequency for a bulk solid is,…”

Section: A Variational Analysismentioning

confidence: 99%

“…40 in which surface elastic constants, including the surface elastic modulus and surface bending modulus, serve as critical parameters influencing the size-dependent behavior of nanowires. 12,13,31,35,38 However, surface elastic constants needed by the surface elasticity theory can only be provided by MD simulation and hardly be measured by proper experiment techniques. 1,13,41,42 How to define the atomic layer as a surface layer, how to choose a proper atomic interaction potential and the size of a numerical model are still open questions in MD simulation.…”

Section: Introductionmentioning

confidence: 99%

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Surface effect on resonant properties of nanowires predicted by an elastic theory for nanomaterials

Yao

Chen

2015

Journal of Applied Physics

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Articles you may be interested inA recently developed continuum theory considering surface effect in nanomaterials is adopted to investigate the resonant properties of nanowires with different boundary conditions in the present paper. The main feature of the adopted theory is that the surface effect in nanomaterials is characterized by the surface energy density of the corresponding bulk materials and the surface relaxation parameter in nanoscale. Based on a fixed-fixed beam model and a cantilever one, the governing equation of resonant frequency for corresponding nanowires is obtained. Numerical calculation of the fundamental resonant frequency is carried out, the result of which is well consistent with the existing numerical ones. Comparing to the result predicted by the conventionally structural dynamics, the resonant frequency of a fixed-fixed nanowire is improved, while that of a cantilever nanowire is weakened due to the surface effect. Both a decreasing characteristic size (height or diameter) and an increasing aspect ratio could further enhance the varying trend of resonant properties for both kinds of nanowires. The present result should be helpful for the design of nano-devices and nanostructures related to nanowires. V C 2015 AIP Publishing LLC.

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Section: Resultsmentioning

confidence: 99%

Section: A Variational Analysismentioning

confidence: 99%

Section: A Variational Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Surface effect on resonant properties of nanowires predicted by an elastic theory for nanomaterials

Yao

Chen

2015

Journal of Applied Physics

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“…As we will show, the fact that the first and second modes show the same trend with decreasing h enables us to distinguish between surface stress and surface elasticity models. First we will discuss a recently proposed model 21 which assumes a distributed transverse force on the cantilever. A strain-independent surface stress ͑ ͒ is introduced into the Euler-Bernoulli equation to account for this force.…”

Section: ͑1͒mentioning

confidence: 99%

Size-dependent effective Young’s modulus of silicon nitride cantilevers

Gavan

Westra

Drift

et al. 2009

Applied Physics Letters

134

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The effective Young's modulus of silicon nitride cantilevers is determined for thicknesses in the range of 20-684 nm by measuring resonance frequencies from thermal noise spectra. A significant deviation from the bulk value is observed for cantilevers thinner than 150 nm. To explain the observations we have compared the thickness dependence of the effective Young's modulus for the first and second flexural resonance mode and measured the static curvature profiles of the cantilevers. We conclude that surface stress cannot explain the observed behavior. A surface elasticity model fits the experimental data consistently. © 2009 American Institute of Physics. ͓DOI: 10.1063/1.3152772͔ Micro-and nanoelectromechanical systems are widely studied for their application in sensing and actuation devices. 1 Down-scaling of such devices improves their sensitivity however at the same time mechanical properties may start to deviate from the bulk behavior. The finite-size effects have been the subject of theoretical studies for the past years. [2][3][4][5][6][7][8] In experimental work on single-crystalline Si cantilevers it has been shown that the Young's modulus strongly depends on the thickness. 9 This behavior has also been observed for suspended crystalline silver nanowires. 10 In describing the properties of nanoscale devices, the bulk Young's modulus ͑E͒ is generally replaced by the effective Young's modulus ͑E eff ͒ to account for size-dependent effects, including surface stress. The total surface stress ͑⌺͒ can be written as a sum of a strain-independent part ͑ ͒ and a strain-dependent part ͑strain ⑀͒, which is related to surface elasticity ͑C s ͒ ⌺ = + C s ⑀. [11][12][13][14][15] In this letter, we study the size-dependency of the Young's modulus in silicon nitride cantilevers when one dimension ͑cantilever thickness͒ is scaled down from 684 to 20 nm. As the SiN x is amorphous, it is difficult to distinguish between the two contributions since parameters ͑e.g., C s ͒ are unknown and difficult to calculate. However, by comparing the experimental data for the first and second mode to theory, we show that the straindependent part of the total surface stress is responsible for the size-dependency.Cantilevers are fabricated from low-pressure chemical vapor deposited ͑LPCVD͒ silicon nitride ͑SiN x ͒ on Si͑100͒ substrates and are patterned with an electron-beam pattern generator. After resist development we use reactive ion etching in a CHF 3 / O 2 ͑20:1͒ plasma to transfer the pattern to the SiN x layer. After this step cantilevers are released using a KOH etch process ͑15 min etching time at 85°C; Si etch rate about 1 m / min͒, yielding facetting along the ͑111͒ planes, as shown in Fig. 1͑a͒. This process introduces a negligible undercut, so that length corrections can be disregarded. 16 Cantilevers are fabricated with the following dimensions: lengths ͑L͒ from 8 to 100 m, widths ͑w͒ 8, 12, or 17 m, and thicknesses ͑h͒ ranging from 20 to 684 nm.The thickness was measured using an ellipsometer ͑Leitz SP͒ with an accuracy of ...

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2012

Carbon Nanotubes and Nanosensors

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Surface stress effect on bending resonance of nanowires with different boundary conditions

Cited by 174 publications

References 20 publications

Surface effect on resonant properties of nanowires predicted by an elastic theory for nanomaterials

Surface effect on resonant properties of nanowires predicted by an elastic theory for nanomaterials

Size-dependent effective Young’s modulus of silicon nitride cantilevers

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