Effect of Surface Stress on the Stiffness of Cantilever Plates

Lachut, Michael J.; Sader, John E.

doi:10.1103/physrevlett.99.206102

Cited by 163 publications

(171 citation statements)

References 20 publications

Supporting

Mentioning

165

Contrasting

Order By: Relevance

“…Changes in the resonant frequency correlating to a change in the surface stress on a cantilever have been observed; 17 however, the changes are small compared with the frequency shift caused by the mass loading effect that a liquid such as water has on a beam and the mechanism driving these changes is still debated. 18 As will be discussed in Sec. II, the thermal noise method and Sader's method are fundamentally different in ways that yield different strengths and weaknesses.…”

Section: Introductionmentioning

confidence: 95%

A method for atomic force microscopy cantilever stiffness calibration under heavy fluid loading

Kennedy

Cole

Clark

2009

Review of Scientific Instruments

View full text Add to dashboard Cite

This work presents a method for force calibration of rectangular atomic force microscopy ͑AFM͒ microcantilevers under heavy fluid loading. Theoretical modeling of the thermal response of microcantilevers is discussed including a fluid-structure interaction model of the cantilever-fluid system that incorporates the results of the fluctuation-dissipation theorem. This model is curve fit to the measured thermal response of a cantilever in de-ionized water and a cost function is used to quantify the difference between the theoretical model and measured data. The curve fit is performed in a way that restricts the search space to parameters that reflect heavy fluid loading conditions. The resulting fitting parameters are used to calibrate the cantilever. For comparison, cantilevers are calibrated using Sader's method in air and the thermal noise method in both air and water. For a set of eight cantilevers ranging in stiffness from 0.050 to 5.8 N/m, the maximum difference between Sader's calibration performed in air and the new method performed in water was 9.4%. A set of three cantilevers that violate the aspect ratio assumption associated with the fluid loading model ͑length-to-width ratios less than 3.5͒ ranged in stiffness from 0.85 to 4.7 N/m and yielded differences as high as 17.8%.

show abstract

Section: Introductionmentioning

confidence: 95%

A method for atomic force microscopy cantilever stiffness calibration under heavy fluid loading

Kennedy

Cole

Clark

2009

Review of Scientific Instruments

View full text Add to dashboard Cite

show abstract

“…2 In principle differential stress in the cantilevers could also affect the effective Young's modulus. 22 This stress may be built-up during the deposition process, when the composition is not homogeneous across the thickness. The resonance frequency of cantilevers is known to be dependent on such stress and the net effect may vary with cantilever thickness.…”

Section: ͑1͒mentioning

confidence: 99%

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

Gavan

Westra

Drift

et al. 2009

Applied Physics Letters

127

View full text Add to dashboard Cite

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 ...

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

Effect of Surface Stress on the Stiffness of Cantilever Plates

Cited by 163 publications

References 20 publications

A method for atomic force microscopy cantilever stiffness calibration under heavy fluid loading

A method for atomic force microscopy cantilever stiffness calibration under heavy fluid loading

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

Contact Info

Product

Resources

About