Surface stress and the normal mode of vibration of thin crystals :GaAs

Łagowski, J.; Gatos, H. C.; Sproles, E.S.

doi:10.1063/1.88231

Cited by 89 publications

(85 citation statements)

References 2 publications

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“…In his model, the effect of residual surface stress is represented by a compressive axial force. Gurtin et al 8 modified this model by applying in addition to the compressive axial force, a distributed traction over the beam surfaces induced by the residual surface tension under bending and concluded that surface elasticity influences the natural frequency of microbeams while residual surface stress does not have any significant effect, in contrast to the results of Lagowski et al 7 The surface/interface tension of fluids can be expressed by the Laplace-Young equation. Gurtin et al 9 formulated a continuum model of surface elasticity in which the LaplaceYoung equation which was extended to solid materials.…”

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confidence: 74%

“…[4][5][6] Therefore, it has been a great theoretical, computational, and experimental activity that has permitted a better understanding of the stress effects on surface physics. Lagowski et al 7 analyzed the natural frequency of microbeams by considering the influence of the residual surface stress on the normal mode of vibration of thin crystals. In his model, the effect of residual surface stress is represented by a compressive axial force.…”

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confidence: 99%

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Free vibration of microscaled Timoshenko beams

Abbasion

Rafsanjani

Avazmohammadi

et al. 2009

Applied Physics Letters

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In this paper, a comprehensive model is presented to investigate the influence of surface elasticity and residual surface tension on the natural frequency of flexural vibrations of microbeams in the presence of rotary inertia and shear deformation effects. An explicit solution is derived for the natural oscillations of microscaled Timoshenko beams considering surface effects. The analytical results are illustrated with numerical examples in which two types of microbeams are configured based on Euler–Bernoulli and Timoshenko beam theory considering surface elasticity and residual surface tension. The natural frequencies of vibration are calculated for selected beam length on the order of nanometer to microns and the results are compared with those corresponding to the classical beam models, emphasizing the differences occurring when the surface effects are significant. It is found that the nondimensional natural frequency of the vibration of micro and nanoscaled beams is size dependent and for limiting case in which the beam length increases, the results tends to the results obtained by classical beam models. This study might be helpful for the design of high-precision measurement devices such as chemical and biological sensors.

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confidence: 74%

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confidence: 99%

Free vibration of microscaled Timoshenko beams

Abbasion

Rafsanjani

Avazmohammadi

et al. 2009

Applied Physics Letters

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“…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.…”

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confidence: 99%

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

Gavan

Westra

Drift

et al. 2009

Applied Physics Letters

132

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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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“…By solving this inverse problem, we actually provide a viable experimental method to determine the effects of both surface elasticity and surface stress. Because the surface stress effect is modelled as an axial load on the structure, two different models arise: the concentrated load model [3][4][5][6][7][8]13,16,24,[35][36][37] and the distributed load model [35][36][37]. The difference between these two models is also discussed.…”

Section: Introductionmentioning

confidence: 99%

“…By the thermodynamics definition, τ is a tensor associated with the reversible work to elastically stretch a pre-existing surface [15]. We see that τ consists of two parts: σ and C s ; σ , which is strain independent, is often referred to as surface stress [16][17][18]; C s , which is strain dependent, is often referred to as surface elasticity [17][18][19]. Surface elasticity is due to the formation of a surface layer that has a different elastic surface stress have different impacts on different resonant frequencies.…”

Section: Introductionmentioning

confidence: 99%

Determining the effects of surface elasticity and surface stress by measuring the shifts of resonant frequencies

2013

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Both surface elasticity and surface stress can result in changes of resonant frequencies of a micro/nanostructure. There are infinite combinations of surface elasticity and surface stress that can cause the same variation for one resonant frequency. However, as shown in this study, there is only one combination resulting in the same variations for two resonant frequencies, which thus provides an efficient and practical method of determining the effects of both surface elasticity and surface stress other than an atomistic simulation. The errors caused by the different models of surface stress and mode shape change due to axial loading are also discussed.

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Surface stress and the normal mode of vibration of thin crystals :GaAs

Cited by 89 publications

References 2 publications

Free vibration of microscaled Timoshenko beams

Free vibration of microscaled Timoshenko beams

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

Determining the effects of surface elasticity and surface stress by measuring the shifts of resonant frequencies

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