Influence of van der Waals force on the pull-in parameters of cantilever type nanoscale electrostatic actuators

Abstract

In this paper, the influence of the van der Waals force on two main parameters describing an instability point of cantilever type nanomechanical switches, which are the pull-in voltage and deflection are investigated by using a distributed parameter model. The fringing field effect is also taken into account. The nonlinear differential equation of the model is transformed into the integral form by using the Green’s function of the cantilever beam. The integral equation is solved analytically by assuming an appropriate shape function for the beam deflection. The detachment length and the minimum initial gap of the cantilever type switches are given, which are the basic design parameters for NEMS switches. The pull-in parameters of micromechanical electrostatic actuators are also investigated as a special case of our study by neglecting the van der Waals force.

Appendix

Solutions of the integrals used in Eq. 29 are

$${\int {\frac{{z^{2}}}{{(a^{2} - z^{2})^{2}}}}}{\text{d}}z = \frac{1}{{4a}}\ln {\left| {\frac{{z - a}}{{z + a}}} \right|} + \frac{z}{{2(a^{2} - z^{2})}}$$

(A.1)

$${\int {\frac{{z^{3}}}{{(a^{2} - z^{2})^{2}}}}}{\text{d}}z = \frac{1}{2}\ln {\left| {z^{2} - a^{2}} \right|} + \frac{{a^{2}}}{{2(a^{2} - z^{2})}}$$

(A.2)

$${\int {\frac{{z^{2}}}{{(a^{2} - z^{2})^{3}}}}}{\text{d}}z = \frac{1}{{16a^{3}}}\ln {\left| {\frac{{z - a}}{{z + a}}} \right|} + \frac{{z(a^{2} + z^{2})}}{{8a^{2} (a^{2} - z^{2})^{2}}}$$

(A.3)

$${\int {\frac{{z^{3}}}{{(a^{2} - z^{2})^{3}}}}}{\text{d}}z = \frac{{2z^{2} - a^{2}}}{{4(a^{2} - z^{2})^{2}}}$$

(A.4)

$${\int {\frac{{z^{2}}}{{a^{2} - z^{2}}}}}{\text{d}}z = - z - \frac{1}{2}a\,\ln {\left| {\frac{{z - a}}{{z + a}}} \right|}$$

(A.5)

$${\int {\frac{{z^{3}}}{{a^{2} - z^{2}}}}}{\text{d}}z = - \frac{1}{2}z^{2} - \frac{1}{2}a^{2} \,\ln {\left| {z^{2} - a^{2}} \right|}.$$

(A.6)