Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-03T00:45:47.841Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-linear effects on canonical MEMS models

Published online by Cambridge University Press:  06 May 2011

NICHOLAS D. BRUBAKER  and
JOHN A. PELESKO
NICHOLAS D. BRUBAKER
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA email: [email protected], [email protected]
JOHN A. PELESKO
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA email: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In modelling electrostatically actuated micro- and nano-electromechanical systems, researchers have typically relied on a small-aspect ratio to form a leading-order theory. In doing so, small gradient terms are dropped. Although this approximation has been fruitful, its consequences have not been investigated. Here, this approximation is re-examined, and a new theory which includes often neglected small curvature terms is presented. Furthermore, the solution set of the new theory is explored for the unit disk domain and compared to the standard theory. Also, the analytical results are compared to experimental data.

Keywords

Prescribed mean curvatureNon-linear eigenvalue problemMircoelectromechanical systemsFold point
Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 5 , October 2011 , pp. 455 - 470
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ackerberg, R. C. (1969) On a nonlinear differential equation of electrohydrodynamics. Proc. R. Soc. A 312 (1508), 129–140.Google Scholar
[2]Burns, M. & Grinfeld, M. (2005) Steady state solutions of a bi-stable quasi-linear equation with saturating flux. Eur. J. Appl. Math., Available at CJO, doi:10.1017/S0956792511000076.CrossRefGoogle Scholar
[3]Esposito, P., Ghoussoub, N. & Guo, Y. (2010) Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS. Courant Lecture Notes. 20, 318. AMS Bookstore.CrossRefGoogle Scholar
[4]Feynman, R. P. (1959) There's plenty of room at the bottom. In: American Physical Society Meeting, Pasadena, CA, USA. Reprinted in J. Microelectromech. Syst. 1, 60–66.Google Scholar
[5]Finn, R. (1986) Equilibrium Capillary Surfaces, Springer-Verlag.CrossRefGoogle Scholar
[6]Ghoussoub, N. & Guo, Y. (2007) On the partial differential equations of electrostatic mems devices: Stationary case. SIAM J. Math. Anal. 38 (5), 1423–1449.CrossRefGoogle Scholar
[7]Gidas, B., Ni, W. M. & Nirenberg, L. (1979) Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68 (3), 209–243.CrossRefGoogle Scholar
[8]Guo, Y., Pan, Z. & Ward, M. J. (2005) Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties. SIAM J. Math. Anal. 66 309–338.CrossRefGoogle Scholar
[9]Joseph, D. D. & Lundgren, T. S. (1973) Quasilinear dirichlet problems driven by positive sources. Arch. Ration. Mech. Anal. 49, 241–269, doi:10.1007/BF00250508.CrossRefGoogle Scholar
[10]Lindsay, A. E. & Ward, M. J. (2008) Asymptotics of some nonlinear eigenvalue problems for a MEMS capacitor. Part I: Fold point asymptotics. Methods Appl. Anal. 15 (3), 297–326.CrossRefGoogle Scholar
[11]Lindsay, A. E. & Ward, M. J. (2011) Asymptotics of some nonlinear eigenvalue problems modelling a MEMS Capacitor. Part II: Multiple solutions and singular asymptotics. Eur. J. Appl. Math. 22, 83–123.CrossRefGoogle Scholar
[12]Nathanson, H. C., Newell, W. E., Wickstrom, R. A. & Davis, J. R. Jr., (1967) The resonant gate transistor. IEEE Trans. Electron Devices 14 (3), 117–133.CrossRefGoogle Scholar
[13]Pan, H. (2009) One-dimensional prescribed mean curvature equation with exponential nonlinearity. Nonlinear Anal. Theory Methods Appl. 70 (2), 999–1010.CrossRefGoogle Scholar
[14]Pelesko, J. A. (2001) Electrostatic field approximations and implications for MEMS devices. In: Proceedings of ESA, pp. 126–137.Google Scholar
[15]Pelesko, J. A. (2002) Mathematical modeling of electrostatic MEMS with tailored dielectric properties. SIAM J. Appl. Math. 62 (3), 888–908.CrossRefGoogle Scholar
[16]Pelesko, J. A. & Bernstein, D. H. (2003) Modeling MEMS and NEMS. CRC Press.Google Scholar
[17]Pelesko, J. A., Bernstein, D. H. & McCuan, J. (2003) Symmetry and symmetry breaking in electrostatically actuated MEMS. Nanotechnology 2, 432–435.Google Scholar
[18]Pelesko, J. A. & Chen, X. Y. (2003) Electrostatic deflections of circular elastic membranes. J. Electrost. 57 (1), 1–12.CrossRefGoogle Scholar
[19]Pelesko, J. A. & Driscoll, T. A. (2005) The effect of the small-aspect-ratio approximation on canonical electrostatic MEMS models. J. Eng. Math. 53 (3), 239–252.CrossRefGoogle Scholar
[20]Pucci, P. & Serrin, J. (2007) The Maximum Principle, Birkhauser.CrossRefGoogle Scholar
[21]Siddique, J. I., Deaton, R., Sabo, E. & Pelesko, J. A. (2011) An experimental investigation of the theory of electrostatic deflections. J. Electrost. 69 (1), 1–6.CrossRefGoogle Scholar
[22]Taylor, G. I. (1968) The coalescence of closely spaced drops when they are at different electric potentials. Proc. R. Soc. A 306 (1487), 423–434.Google Scholar