## Abstract

In this paper, the snap-through buckling of an initially curved microbeam subject to an electrostatic force, accounting for fringing field effect, is investigated. The general governing equations of the curved microbeam are developed using Euler–Bernoulli beam theory and used to develop a new criterion for the snap-through buckling of that beam. The size effect of the microbeam is accounted for using the modified couple stress theory, and intermolecular effects, such as van der Waals and Casimir forces, are also included in our snap-through formulations. The snap-through governing equations are solved using Galerkin decomposition of the deflection. The results of our work enable us to carefully characterize the snap-through behaviour of the initially curved microbeam. They further reveal the significant effect of the beam size, and to a much lesser extent, the effect of fringing field and intermolecular forces, upon the snap-through criterion for the curved beam.

## 1. Introduction

Micro-electro-mechanical systems (MEMS) have aroused great interest for their unique advantages such as small size, high precision and low power consumption. One benchmark of MEMS is the initially straight microbeam system driven by electrostatic force, whose static and dynamic behaviours have been largely investigated in the literature [1–4]. Recently, the bistable MEMS based on initially curved microbeams have drawn more and more attention from the scientific community for their various potential applications such as optical switches, micro-valves and non-volatile memories [5–7].

The initially curved beam (arch) under transverse force may exhibit bistability, and the transition between two stable states is referred to as snap-through buckling [8]. The existence of snap-through buckling depends on various factors, e.g. initial arch rise, beam thickness and clamping angle. Pippard [9] conducted experiments to develop a phase diagram of instability in terms of the arch span and the initial angle at the clamped ends. This work was followed by Patricio *et al.* [10] in which they developed theoretical model simulations to derive a similar phase diagram. As a result of the earlier experiments and model simulations, Krylov *et al.* [11] revealed that the snap-through buckling occurs at large initial deflections. Pane & Asano [12] conducted energy analysis and further found that the existence of bistable states in an initially curved beam depends on the ratio of its initial deflection to its thickness. Park & Hah [13] conducted theoretical investigations and showed that the existence of bistable states also depends on the residual axial stress in the beam. Das & Batra [14] developed a finite-element model to study the transient snap-through behaviour of the initially curved beam, and found that at high loading rates (i.e. voltage is applied at a high rate), the snap-through buckling is suppressed. Moghimi Zand [15] also developed a finite-element model and found the significant inertia effect on the dynamic snap-through behaviour. Medina *et al.* [8,16] examined the symmetric buckling and antisymmetrical bifurcation of electrostatically actuated and initially curved microbeams with/without residual stress. They derived the criteria of symmetric and non-symmetric snap-through instability for quasi-static loading conditions.

Careful literature review indicates that many studies consider a uniform mechanical force as the applied load. However, the electrostatic force applied on the curved microbeam is highly non-uniform and strongly depends on the beam deflection. Several studies consider the electrostatic force, but they fail to examine the fringing field effect and/or the influence of the intermolecular forces such as Casimir and van der Waals forces. Furthermore, the size effect at the microscale is neglected in almost all the existing studies.

The size effect on the deformation behaviour of microstructures has already been observed experimentally [17–20], and such size dependency is attributed to the non-local effects, which cannot be described by the classical continuum theories of local character. Various non-classical continuum theories with additional material length-scale parameters have been proposed [19,21–25]. Among them, the modified couple stress theory developed by Yang *et al.* [25] with a length-scale parameter is one of the most used. Determining the microstructure-dependent length-scale parameters is difficult, so it is desirable to use the theories with only one length-scale parameter [26]. Based on the non-classical continuum theories, the size effect on various behaviours of microbeams has been theoretically studied, including bending, buckling, free vibration, pull-in instability, etc. [27–35].

In this paper, we extend the earlier studies to investigate the size effect on the snap-through behaviour of the initially curved microbeam under electrostatic force. The modified couple stress theory [25] is used. The fringing field effect is taken into account by Meijs–Fokkema formula [36]. The influence of the intermolecular forces is also examined. Based on our model and simulation results, a unified criterion for the existence of snap-through buckling is derived and plotted in a diagram, which can be used for the design of the bistable MEMS.

## 2. Formulation

### (a) General governing equations and boundary conditions

Consider an initially curved rectangular microbeam of span *L*, width *b* and thickness *h* undergoing in-plane bending (*x*–*z* plane in figure 1). The respective displacements *u*_{x}, *u*_{y} and *u*_{z} along *x*-, *y*- and *z*-coordinate are assumed to be dependent only on *x* and *z*. *u*_{y} is further assumed to be 0. For a thin beam (*h*≪*L*), the Euler–Bernoulli beam theory is applied
*a*
*b*
*c*

where *u*(*x*) and *w*(*x*) are, respectively, the axial (along *x*-coordinate) and transverse (along *z*-coordinate) displacements of a point on the mid-plane of the beam; *θ*(*x*) is the rotation of the cross section around *y*-coordinate; a superimposed apostrophe denotes a derivative with respect to *x*. During the snap-through buckling, the mid-plane stretching can be important. To consider this effect, the von Karman nonlinear strain is used. With the aid of equation (2.1), the non-zero strain component (i.e. axial strain *w*_{0}(*x*) by: *ε*_{xx} from equation (2.2) as
*w*_{0}, we obtain the curvature change *χ*_{xy} and *χ*_{yx} from equation (2.5) as
*δU*_{elas} and *δW*_{ext} are, respectively, the variations of the elastic strain energy, and the work done by the external forces. Considering the non-zero strain component *ε*_{xx} and the non-zero curvature components *χ*_{xy} and *χ*_{yx}, we can calculate *δU*_{elas} as [25]
*y*–*z* plane in figure 1). Introduce equations (2.3) and (2.6) into equation. (2.8), integrate the resulting equation by parts with respect to *x*, and we obtain
*N*, *M* and *C* are defined as
*a*
*b*
*c*The variation *δW*_{ext} of the work done by the external forces is
*f*_{x}, *f*_{z} and *m*_{y} are, respectively, the distributed axial load (along *x*-coordinate), transverse load (along *z*-coordinate) and body couple (around *y*-coordinate) per unit length; *N*_{1}, *N*_{2}, *T*_{1}, *T*_{2}, *M*_{1} and *M*_{2} are, respectively, the normal forces, transverse forces and bending moments at the two ends of the beam (i.e. ‘1’ at *x*=0, ‘2’ at *x*=*L*). Replace *θ* with equation (2.1c) in equation (2.11), integrate the obtained-equation by parts with respect to *x*, and we obtain
*δu* and *δw*, we obtain the following governing equations
*a*and
*b*with the following boundary conditions when the corresponding displacements are not specified
*a*
*b*
*c*

### (b) Initially curved microbeam actuated by electrostatic force

Consider an initially curved double-clamped microbeam under electrostatic force (figure 2). Neglecting the gravity, we reduce the governing equations (equation (2.14)) to
*a*and
*b*with the following boundary conditions for the double-clamped beam
*a*
*b*
*c*The distributed transverse load *f*_{z} is composed of
*f*_{elec}, *f*_{Casi} and *f*_{VDW} are, respectively, the electrostatic force, Casimir force and van der Waals force per unit length.

The electrostatic force *f*_{elec} per unit length can be calculated using [37,38]
*V* is the applied voltage difference between the beam and the rigid electrode, *C* is the capacitance per unit length of the capacitor composed of the beam and the electrode and *g* is the gap between the beam and the electrode as being
*g*_{0} being the initial gap (i.e. distance between the clamped beam ends and the rigid electrode (figure 2)). For a small gap *g* (≪ beam length), the beam with the electrode can be regarded as a parallel-plate capacitor. To further take into account the fringing fields at the edges of the microbeam, the capacitance *C* is estimated using the Meijs–Fokkema formula [36]
*ε*_{0} (=8.8542×10^{−12} F⋅m^{−1}) is the vacuum permittivity. It is noted that the error of the estimated capacitance using equation (2.21) is within 6% for the microbeam systems satisfying beam-width-to-gap ratio (*b*/*g*) larger than 0.3 and beam-thickness-to-gap ratio (*h*/*g*) smaller than 10 [36]. So to ensure the proper application of equation (2.21), this paper studies only the microbeam systems with the width-to-initial-gap ratio (*b*/*g*_{0}) larger than 0.5 (considering *w*_{max}=0.5*g*_{0} in equation (2.20)) and the thickness-to-initial-gap ratio (*h*/*g*_{0}) smaller than 5 (considering *w*_{min}=−0.5*g*_{0}). Introduce equations (2.20) and (2.21) into equation (2.19), and after several calculations, we obtain [11]
*a*and
*b*where *h* (=1.0546×10^{−34} J×s) is the reduced Planck constant; *c* (=3×10^{8} m⋅s^{−1}) is the speed of light; *A* (=*π*^{2}*kρ*_{1}*ρ*_{2}) is the Hamaker constant, with *k* being the interaction parameter, *ρ*_{1} and *ρ*_{2} being the numbers of atoms per unit volume in the microbeam and the rigid electrode.

Suppose the beam material is elastically isotropic with Young's modulus *E* and Poisson's ratio *ν*. For an Euler–Bernoulli beam undergoing in-plane bending, we consider only the axial stress *σ*_{xx} [26]. Then, the one-dimensional constitutive relation is
*ψ*_{xy} is related to the symmetric curvature *χ*_{xy} by [25]
*l* is a length-scale parameter. With equations (2.3), (2.6), (2.24) and (2.25), equation (2.10) is changed to
*a*
*b*
*c*where *S*(=*bh*) is the cross-sectional area (*y*–*z* plane in figure 2); *I*(=*bh*^{3}/12) is the second moment of area. Introduce equation (2.26) into equation (2.16), we have
*a*and
*b*With equation (2.27a), equation (2.27b) can be reduced to
*N* is constant, so *N* can be estimated as the average value calculated from equation (2.26a) being *N* with the average value, and considering the boundary condition (equation (2.17*a*)) we obtain
*l* has the effect of increasing the effective bending stiffness (*EI*)_{eff}, being
*h* close to *l*), the effective bending stiffness can be as large as (1+(6)/(1+*v*)) (≈5.7 at *ν*=0.27) times the conventional bending stiffness (*EI*), whereas for thick beams (*h*≫*l*), the effective bending stiffness is nearly equal to the conventional one, indicating that the size effect is negligible.

Rewrite equation (2.30) in the following non-dimensional form [8,43]
*a*and
*b*

### (c) Influence of intermolecular forces

In the non-dimensional governing equation (equation (2.32)), we can identify the dimensionless van der Waals force *a*
*b*
*c*where λ_{VDW}, λ_{Casi} and *β*_{v} are, respectively, the van der Waals force parameter, the Casimir force parameter, and the voltage parameter. With the aid of table 1, we can compare λ_{VDW} and λ_{Casi} with *β*_{v} as follows
*a*and
*b*Consider *g*_{0}≈10^{−6} m for the microscale systems and *V* ≈10^{1} V for the order of applied voltage, and with the values of the constants in table 2, we calculate equation (2.35) as
*a*and
*b*With equations (2.34) and (2.36), the force ratios can be estimated as
*a*and
*b*The maximum force ratios are determined by the minimum stable deflection, i.e. deflection at the pull-in instability, which is roughly half gap (*a*and
*b*Equation (2.38) shows that the intermolecular forces (van der Waals and Casimir forces) are negligible with respect to the electrostatic force when studying the snap-through buckling.

### (d) Reduced-order model

In §2*c*, we have proved that the intermolecular forces can be neglected in the study of snap-through buckling. So the governing equation (equation (2.32)) can be further reduced to
*ϕ*_{k} (*k*=1,2,…,*n*) is the *k*th linear undamped eigenmode of the straight beam, and *q*_{k} is its generalized coordinate. For a double-clamped straight beam, we have
*A*_{k} is a constant satisfying _{k} is a frequency parameter satisfying

It is shown in [43] that the numerical simulations of snap-through buckling using *n*≥6 in equation (2.40) are indistinguishable from each other. It is further stated that a reasonably accurate response of the beam can be given by considering only the first mode (*n*=1) [43], which indicates that the first mode approximation of the deflection can capture the characteristics of the snap-through behaviour. So, in order to simplify our study for an analytical snap-through criterion, we decided to make a first mode approximation here. Suppose the dimensionless initial deflection *a*and
*b*where *q*_{1} is the dimensionless midpoint deflection; *q*_{0} (=*r*/*g*_{0}) is the dimensionless initial arch rise, with *r* being the initial arch rise (i.e. initial deflection at the midpoint). Introduce equation (2.42) into equation (2.39), multiply the result by *ϕ*_{1}, and then integrate over the domain [0,1]. Further integrate by parts with respect to *b*_{11}, *s*_{11} and *I*_{1} are
*a*
*b*
*c*The values of *b*_{11} and *s*_{11} are given in table 3. Equation (2.43) describes the evolution of the voltage parameter *β*_{v} with the dimensionless midpoint deflection *q*_{1}, which will be used to study the snap-through behaviour in the following section.

## 3. Results and discussion

### (a) Influence of initial arch rise on snap-through behaviour

Let us consider an electrostatically actuated microbeam system described by the dimensional quantities in table 4, which is obtained from the experiments in [11]. The corresponding non-dimensional quantities can be calculated with the aid of table 1: stretching parameter *α*=17–96 (with *l*/*h*=0–1, *ν*=0.27 [46]), beam width-to-thickness ratio *b*/*h*=12, dimensionless thickness *q*_{0}=0–0.5, voltage parameter *β*_{v}=0–306 (with *l*/*h*=0–1, *E*=160 GPa and *ν*=0.27 [46]). Taking *α*=95.2798 (*l*/*h*=0.04 with the material length-scale parameter *l* having the order of magnitude of 10^{−1} μm for both silicon and polysilicon [47]), *b*/*h*=12, *b*_{11}=198.4626 and *s*_{11}=4.8777 (table 3), we plot equation (2.43) in figure 3 at different levels of *q*_{0} (0–0.5). The experimental results from [11] are also shown. It is seen that the model (equation (2.43)) can approximately describe the snap-through behaviour observed from the experiments. The difference in the critical voltages (i.e. voltage parameters at the extreme points) is possibly owing to the non-ideal clamping conditions, residual stresses, initial imperfections in the beam shape and variations of beam geometry owing to the low fabrication tolerances [11].

Figure 3 shows that the existence of the snap-through buckling depends on the level of the dimensionless initial arch rise *q*_{0}. For very small *q*_{0} (e.g. *q*_{0}=0 in figure 3*a*), there is only one extreme point *q*_{p} on the *β*_{v}−*q*_{1} curve corresponding to the pull-in instability. With the increase of the voltage (*β*_{v} increases), the microbeam bends towards the rigid electrode owing to the electrostatic force. The equilibrium position of the beam can be determined by the balance of the elastic and electrostatic forces. Therefore, the beam deflection decreases gradually (see the loading path A→*q*_{p} in figure 3*a*). When the critical point *q*_{p} is reached, the microbeam becomes unstable (i.e. the elastic force can no longer resist the electrostatic force), so it collapses onto the rigid electrode (*q*_{p}→*B*). This behaviour is called pull-in instability.

For a larger value of *q*_{0} (e.g. 0.35 in figure 3*b*), two more extreme points *q*_{s} and *q*_{r} appear, which correspond, respectively, to the snap-through buckling and the release (snap-back). With the increase of *β*_{v}, the beam deflection decreases gradually (C→*q*_{s} in figure 3*b*) until reaching the critical point *q*_{s} where two stable states (*q*_{s} and D) coexist. A slight increase in *β*_{v} causes a sudden transition from the initial stable state *q*_{s} to the second stable state D, so the beam deflection suddenly decreases (*q*_{s}→D). Such transition is called snap-through buckling. After the snap-through buckling, the beam deflection continues to decrease gradually with *β*_{v} (D→*q*_{p}) until reaching the pull-in instability where the beam collapses onto the rigid electrode (*q*_{p}→E).

If *q*_{0} is large (e.g. 0.48 in figure 3*c*), the voltage parameter *β*_{v} at the snap-through buckling point *q*_{s} is larger than that at the pull-in instability point *q*_{p}. Then, the snap-through and the pull-in take place simultaneously, as shown by the loading path F→*q*_{s}→G in figure 3*c*. In this case, the observed behaviour of the microbeam is similar to that of the ordinary pull-in instability. In summary, to exhibit the snap-through behaviour (separately from the pull-in instability), the microbeam should have a dimensionless initial arch rise *q*_{0} in a certain range.

### (b) Size and fringing field effects on snap-through behaviour

The fringing field effect owing to the finite beam width on the snap-through behaviour is shown in figure 4*a* by plotting equation (2.43) at different levels of width-to-thickness ratio *b*/*h* (*q*_{0}=0.35, *α*=95.2798, *b*_{11}=198.4626 and *s*_{11}=4.8777. It is seen that with the decrease of *b*/*h*, the voltage parameter at the snap-through buckling (point *q*_{s}) decreases. This is due to the fact that when reducing *b*/*h* (beam width decreases), the fringing field effect becomes more significant, which increases the total voltage between the beam and the rigid electrode. Therefore, less applied voltage (normalized as *β*_{v}) is needed to induce the snap-through buckling.

The size effect (considering the length-scale parameter *l*) on the snap-through behaviour is shown in figure 4*b* by taking *q*_{0}=0.35, *b*/*h*=12, *b*_{11}=198.4626 and *s*_{11}=4.8777 in equation (2.43). At different levels of the length-scale parameter *l*/*h*=0–1, the stretching parameter *α* is calculated with the expression and the microbeam system dimensions given in tables 1 and 4. Figure 4*b* shows that the critical points (*q*_{s}, *q*_{r}) of snap-through buckling disappear when *l*/*h* is larger than 0.7, which indicates that *q*_{0}=0.35 is not in the domain of snap-through buckling for thin microbeams with *l*/*h*≥0.7. Size effect influences the domain of the dimensionless initial arch rise *q*_{0} for the snap-through buckling. In §3*c*, we derive an analytical expression of the domain of *q*_{0} (i.e. analytical criterion) for the existence of snap-through buckling.

### (c) Size and fringing field effects on snap-through criterion

The extreme points *q*_{s}, *q*_{r} and *q*_{p} on *β*_{v}−*q*_{1} curve (refer to figure 3) can be obtained by solving the following equation with the aid of equation (2.43)
*I*_{2} is calculated from equation (2.44c) as
*I*_{1}, *I*_{2}) cannot be solved analytically. So we solve the equation numerically, and show the typical results in figure 5. It is seen that *q*_{0} must be larger than a critical value *q*_{s} and *q*_{r} related to the snap-through buckling. At *q*_{s} and *q*_{r} are near 0. So for a first approximation, we take *q*_{1}=0 in equation (3.1) and find

where the expressions and values of *b*_{11} and *s*_{11} are given in table 3; *a*and
*b*

When the dimensionless initial arch rise *q*_{0} exceeds a critical value *β*^{snap-through}_{v} at the snap-through buckling (point *q*_{s}) becomes larger than that *β*^{pull-in}_{v} at the pull-in instability (point *q*_{p}), so the snap-through and the pull-in take place simultaneously (figure 3*c* in §3*a*). To determine the critical value *q*_{s}(*q*_{0}) and *q*_{p}(*q*_{0}) with *q*_{0}. (ii) Take *q*_{1}=*q*_{s}(*q*_{0}) and *q*_{p}(*q*_{0}), respectively, in equation (2.43) to obtain the evolutions of the voltage parameters *β*^{snap-through}_{v}(*q*_{0}) and *β*^{pull-in}_{v}(*q*_{0}) at the extreme points. (iii) Compare *β*^{snap-through}_{v}(*q*_{0}) with *β*^{pull-in}_{v}(*q*_{0}), and determine the critical value *α* and width-to-thickness ratio *b*/*h* is shown in figure 6, from which it is found that *α* (54–2400 with *l*/*h*=0–1) and *b*/*h* (1.5–100). Because the relative variation of *α* and *b*/*h*, and take: *q*_{0} for the existence/observation of snap-through buckling:
*q*_{0} (terms on the left of *q*_{0}) depends on the stretching parameter *α*, which is a function of the length-scale parameter *l* (table 1). So, the size effect is implicitly included in equation (3.5). In the rest of this section, we derive a criterion of snap-through buckling from equation (3.5), which is explicitly expressed in term of *l*.

By introducing equation (3.4) into equation (3.5) and replacing the non-dimensional quantities (*α*, *q*_{0}) with the expressions in table 1, we obtain the following criterion for the existence/observation of snap-through buckling:
*r*/*h*)_{min},(*r*/*h*)_{max}) between the initial arch rise *r* and beam thickness *h* are given by
*a*and
*b*where the values of the constants *b*_{11}, *s*_{11}, *f*_{1} and *m*_{11} are given in table 3;

It is noted that the ratio (*r*/*h*) should also be below a critical value to suppress the asymmetric second mode [11,46]. By comparing with the symmetry breaking criterion [8], we find that for the small gap-to-thickness ratio *g*_{0}/*h* (less than or equal to 4.5), the maximum allowable ratio (*r*/*h*)_{max} from our criterion is smaller than that from the symmetry breaking criterion. So, (*r*/*h*)_{max} of our criterion can be used at the small ratio *g*_{0}/*h* (≤4.5), whereas at large ratio, *g*_{0}/*h* (>4.5), (*r*/*h*)_{max} from the symmetry breaking criterion [8] should be used.

The size effect (by introducing the length-scale parameter *l*, normalized as *l*/*h*) and the fringing field effect (considering the finite beam width *b*, normalized as *b*/*h*) on the minimum allowable ratio (*r*/*h*)_{min} are shown, respectively, in figure 7*a*,*b* from equation (3.7a). Both effects increase (*r*/*h*)_{min} and the size effect is much more significant. Equation (2.31) shows that the size effect (*l*/*h*) increases the effective bending stiffness, so the microbeam becomes stiffer and more difficult to exhibit snap-through buckling. As a result, the minimum allowable ratio (*r*/*h*)_{min} increases.

With *l*/*h*=0.04 and *b*/*h*=12, the snap-through criterion (i.e. equation (3.6): domain of the ratio (*r*/*h*) between the initial arch rise and beam thickness) is plotted in figure 7*c*. It is seen that for the existence of the domain (*r*/*h*) of snap-through buckling, the gap-to-beam-thickness ratio *g*_{0}/*h* should be large enough (e.g. *g*_{0}/*h*>2 in figure 7*c*). The experimental results in the literature [11,46] are also shown in the figure, and it is found that most of the observed snap-through buckling takes place within the predicted area. The snap-through criterion can be used as a design guideline for the bistable MEMS based on the initially curved microbeam: consider a microbeam system of prescribed dimensions (beam thickness *h* and width *b*) and made of prescribed material (silicon, polysilicon, epoxy, etc.), determine the length-scale parameter *l* of the material, and then with the calculated (*b*/*h*) and (*l*/*h*), plot equation (3.6) in a figure similar to figure 7*c*. The initial arch rise *r* of the microbeam and the gap *g*_{0} between the beam ends and the rigid electrode can be chosen in the snap-through area on the figure.

It is noted at last that we can compare the experiments of the two research groups in the same figure (figure 7*c*) by taking the same values of *l*/*h* and *b*/*h* in equation (3.7a), because (i) the microbeams tested in these two groups are made of the same material—silicon, and the order of the material length-scale parameter *l* for silicon is 10^{−1}μ m [47]. Moreover, the beam thicknesses *h* are very close: *h*=2.5 [11] and 2.6 μm [46]. So, we choose one value of *l*/*h*=10^{−1} μm/2.5 μm=0.04. (ii) The width-to-thickness ratios *b*/*h* are also close: *b*/*h*=12 [11] and 19 [46], and it is seen from figure 7*b* that the variation of (*r*/*h*)_{min} over a wide range of *b*/*h* (*b*/*h*=12.

## 4. Conclusions

This paper is concerned with a unified study of the snap-through behaviour of an initially curved microbeam subjected to an electrostatic force and accounting for fringing field and intermolecular effects. The governing equations were developed with the aid of Euler–Bernoulli beam theory and used to develop a new snap-through criterion in terms of the microbeam system dimensions; accounting for the beam size and the fringing field effect in the development of that criterion. The governing equations were solved using the Galerkin decomposition method and used to develop a limit design chart for the characteristic snap-through behaviour of the beam. Our results, which are based on the first mode approximation, reveal that the size of the microbeam plays a major role in dictating the existence of the snap-through behaviour of the beam, whereas the fringing field and intermolecular forces play an insignificant role.

It is noted that to derive an analytical snap-through criterion, the first mode approximation of the beam deflection was taken. So the model can only approximately describe the evolution of the beam deflection with the applied voltage. To accurately describe the deflection evolution, more modes (greater than or equal to 5) should be considered, and some cases such as non-ideal clamping conditions and imperfections in the beam shape should also be taken into account. The derived snap-through criterion is only valid for the microbeam systems with gap-to-beam-thickness ratio smaller than 4.5, whereas for larger ratios, the symmetry breaking criterion [8] with two modes considered should be used. Another limitation is that the work is only valid for long thin beams (beam thickness ≪ beam length) where the Euler–Bernoulli beam theory can be applied. Moreover, the Meijs–Fokkema formula for narrow beams was used to take into account the fringing field effect. To ensure the application of the formula, this work only studied the microbeam systems with the dimensions satisfying beam-width-to-gap ratio larger than 0.5 and beam-thickness-to-gap ratio smaller than 5 (i.e. gap-to-beam-thickness ratio larger than 0.2).

## Data accessibility

There is no data to report in this manuscript.

## Funding statement

The research is funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Discovery Accelerator Supplements (DAS).

## Authors contributions

X.C. carried out the research work and drafted the manuscript. S.A.M. helped correct and improve the manuscript. X.C. and S.A.M. gave final approval for publication.

## Competing interests

We have no competing interests.

## Acknowledgements

The authors thank the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Discovery Accelerator Supplements (DAS) for their financial support of the current studies. They also wish to thank the anonymous reviewers for their constructive comments.

- Received February 2, 2015.
- Accepted March 11, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.