Nonlinearity in nanomechanical cantilevers

L. G. Villanueva, R. B. Karabalin, M. H. Matheny, D. Chi, J. E. Sader, and M. L. Roukes

Phys. Rev. B 87, 024304 – Published 28 January 2013

Abstract

Euler-Bernoulli beam theory is widely used to successfully predict the linear dynamics of micro- and nanocantilever beams. However, its capacity to characterize the nonlinear dynamics of these devices has not yet been rigorously assessed, despite its use in nanoelectromechanical systems development. In this article, we report the first highly controlled measurements of the nonlinear response of nanomechanical cantilevers using an ultralinear detection system. This is performed for an extensive range of devices to probe the validity of Euler-Bernoulli theory in the nonlinear regime. We find that its predictions deviate strongly from our measurements for the nonlinearity of the fundamental flexural mode, which show a systematic dependence on aspect ratio (length/width) together with random scatter. This contrasts with the second mode, which is always found to be in good agreement with theory. These findings underscore the delicate balance between inertial and geometric nonlinear effects in the fundamental mode, and strongly motivate further work to develop theories beyond the Euler-Bernoulli approximation.

Received 27 March 2012

DOI:https://doi.org/10.1103/PhysRevB.87.024304

Authors & Affiliations

L. G. Villanueva^1,2, R. B. Karabalin¹, M. H. Matheny¹, D. Chi¹, J. E. Sader^1,3, and M. L. Roukes¹

¹Kavli Nanoscience Institute and Departments of Physics, Applied Physics, and Bioengineering, California Institute of Technology, Pasadena, California 91125, USA
²Department of Micro- and Nanotechnology, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark
³Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 87, Iss. 2 — 1 January 2013

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Schematic showing the type of device investigated in this study: a U-shaped cantilever beam of total length $L$ and width $b$ . The region close to the anchor presents two legs of length $L_{leg}$ and width $b_{leg}$ .Reuse & Permissions
Figure 2
Schematic for the fabrication process flow. Side view is shown in the left column and the corresponding top view is depicted in the right column. (a) SiN is deposited on both sides of a Si wafer. Back side SiN is patterned to define windows for the subsequent anisotropic silicon etching in KOH, yielding membranes on the front side (b). We then deposit (c) two bilayers Cr/Au by means of two subsequent liftoff processes: one to be used in the detection of motion (5 nm/50 nm Cr/Au) and another one to define the contacts (5 nm/150 nm Cr/Au). (d) Using the gold as a hard mask, we perform a mild dry etching of the silicon nitride layer which defines the released structures with a proper clamping region, i.e., with no undercut (see also Fig. 1).Reuse & Permissions
Figure 3
Examples of nanomechanical cantilever devices, (a) and (b), and their respective nonlinear responses for the first [(c),(d)] and second [(e),(f)] flexural vibrational modes. The micrographs show four structures with different ARs (AR, length over width): 5, 7 (a) and 12, 13 (b). The resonant responses (c)–(f) show the amplitude of vibration as a function of the drive frequency for different magnitudes of the driving force. The data correspond to cantilevers of AR 5 [(c),(e)] and AR 13 [(d),(f)]. Scale bars are 5 μm.Reuse & Permissions
Figure 4
Experimentally estimated nonlinear coefficients for the first (a) and the second (b) out-of-plane vibrational modes. While data for the second mode (b) show good agreement with theory, data for the first mode (a) clearly diverge from the calculated value, mainly for low AR. This is highlighted when plotting the relative difference between experiment and theory (c). The first mode shows around one order of magnitude difference with the expected value for low AR. This difference is reduced for large AR. The second mode experimental measurements lie within some tens of percent of the expected value. A gray solid line represents the theoretical prediction in each plot. In (a) a dotted line delineates softening and stiffening behavior, while a dashed line is presented as a visual aid to follow the systematic experimental trend. In (b) the dashed line represents the average experimental value and the boundaries of the colored zone define one standard deviation from the mean.Reuse & Permissions
Figure 5
Resonant responses for different driving forces for the first [(a),(b)] and second [(c),(d)] mode of two cantilevers with the same AR in the design (AR 7—due to alignment tolerances the real AR is 7.62 [(a)–(c)] and 7.43 [(b)–(d)]. We show that the second mode presents softening nonlinearity on both cases, while the first mode presents both stiffening (a) and softening (b) nonlinearities. This may be due to additional unspecified effects such as material nonlinearity or fabrication-related differences (surface damage, polymeric residues, etc.).Reuse & Permissions
Figure 6
SEM micrographs showing two devices that were designed to have an identical aspect ratio of 7. Slight differences due to alignment mismatch can be observed. The nonlinear response of these devices (a) is presented in Figs. 5a and 5c, while the nonlinear response for device (b) is given in Figs. 5b and 5d.Reuse & Permissions

Physical Review B

covering condensed matter and materials physics