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

Abstract

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 ...

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

“…16) There is no damping in equations (2.7) and (2.12), which is also the case in many studies [3,4,6,7,13,18,19,33]. The presence of damping reduces the resonance, and damping can be determined by the so-called half-power method from the frequency response curve obtained by experiment [45].…”

“…However, Gurtin & Murdoch [51] Clearly because of the positive Δ and N, the beam stiffness is enhanced and the eigenfrequencies thus all increase when compared with those in equation (3.2). In the beam resonance test, Δ and N are unknown; the eigenfrequencies are extracted from the beam frequency response curves [13,16,32,33]. Therefore, using the eigenfrequencies to determine Δ and N forms an inverse problem.…”

“…The concentrated load F is due to the strain-independent surface stress of σ , and F = πσ D for a circular beam (D is the diameter of a circular beam). For a rectangular beam with a width b and a thickness h, F = 2σ b = 2σ A/h (A = bh) [13,16,19], which only includes the load induced by the surface stress on the upper and lower surfaces. We see that ∂ 2 w/∂x 2 is the curvature of the upper/lower surface and F is in the horizontal direction; −F∂ 2 w/∂x 2 is a distributed transverse load given by the Young-Laplace (YL) formula [38].…”

“…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.…”

“…In a surface layer, the total surface stress (τ ) is given as follows [2,4,5,[8][9][10][11][12][13]: 1) where is the dimensionless strain and C s is the surface modulus. Here, τ is the result of charge redistribution as the electrons respond to the effects of terminating a solid at a surface [14].…”