Stress-Induced Variations in the Stiffness of Micro- and Nanocantilever Beams

The effect of surface stress on the stiffness of cantilever beams remains an outstanding problem in the physical sciences. While numerous experimental studies report significant stiffness change due to surface stress, theoretical predictions are unable to rigorously and quantitatively reconcile these observations. In this Letter, we present the first controlled measurements of stress-induced change in cantilever stiffness with commensurate theoretical quantification. Simultaneous measurements are also performed on equivalent clamped-clamped beams. All experimental results are quantitatively and accurately predicted using elasticity theory. We also present conclusive experimental evidence for invalidity of the long-standing and unphysical axial force model, which has been widely applied to interpret measurements using cantilever beams. Our findings will be of value in the development of micro- and nanoscale resonant mechanical sensors.

Figure 1

Resonant response of piezoelectric beams. a, SEM micrograph of the doubly clamped beams used for the experiments. On top of the micrograph, we show resonant responses of each of the beams, yielding resonant frequencies of 38.3 MHz (length $6\mu \mathrm{m}$, purple), 22.9 MHz ($8\mu \mathrm{m}$, green) and 14.8 MHz ($10\mu \mathrm{m}$, blue). Experimental details are provided in 27. b, SEM micrograph of the cantilever beams used for the experiments. Respective resonant responses are also shown for each cantilever, yielding natural frequencies of 8.85 MHz (length $6\mu \mathrm{m}$), 4.82 MHz ($8\mu \mathrm{m}$), 3.16 MHz ($10\mu \mathrm{m}$). Both types of beams have the same composition (320 nm of total thickness) and width (900 nm). Lengths are 6, 8, or $10\mu \mathrm{m}$ for both types of devices, causing the boundary conditions to be the only difference, thus allowing proper comparison of the experimental results for the two configurations. Scale bars: $2\mu \mathrm{m}$.

Figure 2

Frequency shift results for doubly clamped beams. a, Relative frequency shift $\Delta f/f_R$ for the three doubly clamped beams in Fig. 1a, showing the dependence of $\Delta f/f_R$ on length (as predicted by theory). b, Absolute frequency shift $\Delta f$ (in kHz) for the same three beams. Experimental (lighter colors) and FEM results (scattered plots, darker colors) display excellent agreement. The measurements are shown as a function of the applied dc bias (in volts) and as a function of the corresponding surface stress, as calculated in 27. The stress calculation requires estimation of the piezoelectric coefficient $d_{31}$; this was obtained by a linear fit of these experimental results, yielding a value of $-2.5\mathrm{pm}/\mathrm{V}$.

Figure 3

Frequency shift results for cantilever beams. a, Relative frequency shift $\Delta f/f_R$ for the three cantilever beams in Fig. 1b, showing independence of length, which is consistent with the theoretical model. Both experimental (lighter colors) and FEM results (scattered plots, darker colors) are presented. FEM data are a result of combining both stress and geometric effects; these show excellent agreement with measurements. b, Absolute frequency shift $\Delta f$ (in kHz) for the same three beams. Comparison of these results with those in Fig. 2 demonstrates that the effect of stress on the resonant frequency is much smaller for cantilevers than doubly clamped beams.