MEMS and Nanotechnology, Volume 6 pp 67-74 | Cite as

# Stiction Failure in Microswitches Due to Elasto-Plastic Adhesive Contacts

## Abstract

Undesirable stiction, which results from the contact between surfaces, is a major failure mode in micro-switches. Indeed the adhesive forces can become so important that the two surfaces remain permanently glued, limiting the life-time of the MEMS. This is especially true when the contact happens between surfaces where elasto-plastic asperities deform permanently until the surfaces reach plastic accommodation, increasing the surface forces. To predict this behavior, a micro adhesive-contact model is developed, which accounts for the surfaces topography evolutions during elasto-plastic contacts. This model can be used at a higher scale to study the MEMS behavior, and thus its life-time. The MEMS devices studied here are assumed to work in a dry environment. In these operating conditions only the Van der Waals forces have to be considered for adhesion. For illustration purpose, an electrostatic-structural analysis is performed on a micro-switch. To determine the degree of plasticity involved, the impact energy of the movable electrode at pull-in is estimated. Thus the maximal adhesive force is predicted using the developed model.

## 11.1 Introduction

The inherent characters of MEMS such as the large surface area-to-volume ratio, smooth surfaces, small interfacial gaps and small restoring forces, make them particularly vulnerable to stiction which is one of the most common failure mechanism of MEMS [1]. Stiction happens when two components entering into contact permanently adhere to each-other because the restoring forces are smaller than the surface forces (capillary, van der Waals (VDW) or electrostatic). This can happen either during the fabrication process at etching (release stiction) or during normal use (in-use stiction).

To improve the reliability of MEMS, models are required in order to predict and avoid in-use stiction failure. A multi-scale model can predict at the lower scale the adhesive contact forces of two rough surfaces, and thus can integrate these curves on the surface of the finite elements as a contact law at the higher scale [2, 3]. The authors recently proposed [4] a model predicting the micro adhesive-contact curves, i.e. the adhesive-contact force vs. the surface separation distance, for two interacting micro-surfaces. This analytical model, accounting for elastic deformations of the asperities, and for van der Waals forces, is based on classical adhesion theories [5, 6, 7, 8, 9, 10] and can be easily integrated in the multiscale framework [2, 3].

Although the two-scale framework [3] based on the elastic micro-model [4] has been shown to predict accurate results for elastic materials in dry environment [3], in order to extend the applicability of the method to other environments, the micro-model requires enhancements, and in particular its extension to the elasto-plastic behavior of the asperities. As a first step toward this end, this paper presents an improved model for the single elastic–plastic asperity-plane interaction problem.

When elastic–plastic rough surfaces interact, each asperity will be affected differently due to the statistical nature of the asperity distribution on the surfaces: higher asperities will experience plastic deformations first. Due to the plastic behavior, the contact force on deformed asperities is lower than in the elastic case for the same contact interference (distance between undeformed profiles), while the adhesive force increases due to the change of the asperity profile. Because of the combination of these two phenomena the pull-out force – maximum attractive forces or the minimum compressive forces between the two interacting surfaces – is higher than that between two pure elastic contacting rough surfaces. Another qualitative difference with elastic surfaces is the difference of behavior under cyclic loading: after repeated contacts, the distribution of asperities heights and the tip radii of the higher asperities change [11], until plastic accommodation or shakedown [11]. This induces a “contact hardening” [13] and the pull-out force increases until accommodation, unless in-use stiction happens first.

To account for the elasto-plastic behavior, the authors [14] have developed a micro-model able to predict stiction for elastic–plastic rough surfaces by first considering the problem of a single elasto-plastic asperity interaction and thus the generalization to the interaction of rough surfaces. The single asperity/plane contact problem is modeled using semi-analytical models [15, 16, 17] which evaluate the deformed asperity profile during hysteretic loading/unloading without considering the adhesion effect. Assuming adhesion will not affect the plastic deformations, which is not the case for extremely soft materials as gold [18], we can consider the Maugis theory [7] completed by Kim expansion [8] to evaluate the adhesion forces during the unloading phase [4] from the tip radius evolution during loading process. As a main difference with previous models [15, 16, 17], adhesion forces are evaluated taking into account the effect of the non-constant asperity curvature resulting from elasto-plastic deformations, which conducts to an accurate prediction of the pull-out forces [14]. In this model only van der Waals forces are considered, which is a realistic assumption below 30% humidity [1]. The interaction of two rough surfaces is achieved by considering a usual statistical distribution of asperities [5, 6], however, contrarily to the elastic case, the distribution of asperities heights and the asperity profiles of the higher asperities change due to the plastic deformations. These changes, and the resulting adhesive-contact forces, are evaluated using the single asperity model. As a result, micro adhesive-contact curves of two interacting elasto-plastic rough surfaces can be predicted in an analytical way during loading and unloading.

The purpose of this paper is to predict the reliability of a micro-switch by considering the effect of repeated interactions between the movable/substrate electrodes. For illustration purpose a one-dimensional model is considered and contact occurs between two Ruthenium (Ru) films. We also show that unloading curves change after repeated interactions until reaching accommodation. Thus, the pull-out force can be predicted in terms of the pull-in force and of the cycles number, opening the way to a stiction-free design.

The organization of the paper is as follows. In Sect. 11.2, the micro-model for elasto-plastic adhesive-contact is summarized. First the single elasto-plastic asperity/plane interaction model with no adhesion effect is described. Then, the adhesion forces are evaluated from the deformed asperity profile taking into account the effect of the non-constant asperity curvature resulting from elasto-plastic deformations. Finally the micro adhesive-contact curves of two interacting elasto-plastic rough surfaces are deduced. This model can then be used in Sect. 11.3 to study the micro-switch reliability. In particular the effect of cyclic loading on the pull-out force, and thus on the stiction risk, is predicted.

## 11.2 Micro-model for Elasto-Plastic Adhesive-Contacts

In this section, the single elasto-plastic asperity/plane interaction model with no adhesion effect is first described before evaluating the adhesion forces from the deformed asperity profile. Then, using a statistical distribution of asperities heights accounting for the changes in asperity profiles and heights due to the plastic deformations, the micro adhesive-contact curves of two interacting elasto-plastic rough surfaces can be predicted.

### 11.2.1 Single Asperity Elasto-Plastic Contacts

*R*, Young modulus

*E*, and yield stress

*S*

_{Y}, interacts with a rigid plane at an interference distance

*δ*, positive in case of contact and negative otherwise, see Figs. 11.1a, b, defined as the distance between the original profile of the asperity tip and the plane. When the plane starts interacting with the asperity during loading, the critical yield interference

*δ*

_{CP}is defined as the interference at which the asperity starts yielding and can be expressed as [15, 16, 17]

*C*

_{v}is a coefficient that depends on the Poisson ratio

*v*, and that can be evaluated from

*C*

_{v}

*=*1.295e

^{0.736v}, e.g. [16]. As, with our assumption, the asperity starts yielding at positive interference, there exists a corresponding critical contact radius

*a*

_{CP}, Fig. 11.1a, and a critical contact force

*F*

_{CP}, respectively evaluated as

*δ*

_{CP}, the asperity is subject to permanent plastic deformations that depend on

*δ*

_{max}, the maximal interference reached. After unloading, the asperity exhibits a permanent reduction of the asperity height

*δ*

_{res}, and a modified asperity tip radius

*R*

_{res}, see Fig. 11.1c, that were curve-fitted from finite element numerical simulations [17]

### 11.2.2 Single Asperity Elasto-Plastic Adhesive Contacts

*z*

_{0}, two surfaces are attracted with a constant force per unit area σ

_{0}, while if the separation

*z*exceeds

*z*

_{0}, the adhesive traction vanishes. The associated adhesive energy reads ϖ = σ

_{0}

*z*

_{0}. Maugis theory for the interaction of two elastic asperities characterized by two Young modulii

*E*

_{1}and

*E*

_{2}, two Poisson ratios

*v*

_{1}and

*v*

_{2}, and by two tip radii

*R*

_{1}and

*R*

_{2}, is based on the definition of a transition parameter

*δ*, the adhesive contact force

*F*

_{n}, the interacting contact radius

*a*and the adhesive-contact radius

*c*on which adhesive forces apply, see Fig. 11.1a. The system of equations is written in terms of the non-dimensional values

Kim et al. [8] extended Maugis-Dugdale solution to the non-contact regime when *a* = 0 and c ≠ 0, see Fig. 11.1b, see [4] for details. Practically, this expansion has to be considered when λ <0.938.

*R*

_{eff}at a contact interference δ −

*δ*

_{res}. This is motivated by the fact that Maugis theory assumes a uniform asperity radius to apply Hertz theory although this case is only met at the limit case

*δ*=

*δ*

_{res}. The following expression has been proposed [14]

*c*

_{1}and

*c*

_{2}are functions that have to be determined by inverse analysis from finite-element results. Using the simulations performed for Ru [19], we proposed [14]

Because of the elasto-plastic behavior happening during contacts, the theory developed here results in different adhesive-contact forces during loading *F*_{n}^{L}(*δ*) and unloading *F*_{n}^{U}(*δ*). During the loading phase, once δ_{CP} is reached, the maximum interference is identical to the current one (*δ*_{max} = *δ*) and the deformed profile can be evaluated from (11.4) and (11.5). Thus, the loading force *F*_{n}^{L}(*δ*) is evaluated from Maugis solution by solving the system (11.8), (11.9), (11.10), and (11.11), with as input for *R* the effective radius (11.12), and as input for *δ* the effective value *δ* − *δ*_{res}, where *δ*_{res} increases during the whole loading process. During unloading however, the residual (*δ*_{res}) and maximal (*δ*_{max}) interferences reached remain constant. The adhesive-contact force during unloading *F*_{n}^{U}(*δ*) is computed from the Kim extension [8] of Maugis theory [7], with as input for *R* the effective radius (11.12), and as input for *δ* the effective value *δ* − *δ*_{res}. Contrarily to the loading process, the effect of adhesion needs to be considered at the intermediate pull-out stage, which is achieved by using the Kim extension [8].

*δ*

_{max}successively equal to 17, 34 and 51 nm. It is seen that an excellent agreement is obtained for the three loading conditions.

Properties of Ru films

| 4 |

| 410 |

| 0.3 |

| 3.42 |

| 0.169 |

ϖ [J/m | 1 |

σ | 7.78 |

| 7.81 |

N [μm | 10 |

### 11.2.3 Rough Surfaces Interaction

*R*, whose heights

*h*have a statistical distribution

_{s}is the standard deviations in asperity heights. The contact of two rough surfaces can be represented by the contact between an equivalent rough surface and a smooth plane [10]: if the two initial contacting rough surfaces have respectively the asperities end radii of

*R*

_{1}and

*R*

_{2}, the equivalent radius is defined by (11.6), and if the standard deviation in asperity heights are

*σ*

_{s1}and

*σ*

_{s2}, the equivalent rough surface is defined by the standard deviation in asperity heights

*σ*

_{s}= (

*σ*

_{s1}

^{2}+

*σ*

_{s2}

^{2})

^{1/2}. The interaction between two rough surfaces is also characterized by the distance

*d*between the two rough surface mean planes of asperity heights, and by

*N*, the surface density of asperities. All these values can be identified from the study of the surfaces topography, and, in particular, depend on the surface RMS roughness

*R*

_{q}, see [3] for details.

*F*

_{nT}

^{L}and

*F*

_{nT}

^{U}can now be evaluated by integrating on the surface the effect of each asperity, for which the interference reads

*δ*=

*h − d*, using the framework described in Sect. 11.2.2. Toward this end, non-dimensional values are defined

## 11.3 Cyclic Loading of a Micro-switch

*U*is applied between a movable electrode and a substrate electrode covered by a dielectric layer of thickness

*t*

_{d}and permittivity ε

_{d}. The movable electrode is attached to a spring of stiffness per unit area

*K*

_{S}, and is initially at a distance

*d*

_{0}from the substrate. The switch is supposed to work in vacuum, permittivity

*ε*

_{0}, so the damping effect of a squeeze film is neglected. Typical values for SiN dielectric are reported in Table 11.2. Contact is assumed to occur between two Ru surfaces, for which typical topography values are reported in Table 11.1. Ru films of thickness

*t*

_{s}are deposited on the movable electrode, and also on a part of the substrate.

Properties of the micro-switch

| 2 |

| 0.15 |

| 8.854 |

| 7.6 |

| 180 |

*K*

_{S}. This computation has been performed in [14], and in this application we consider an impact energy of

*E*

_{I}= 0.5 J/m

^{2}. This impact energy per unit surface of the contacting area affects the plastic deformations of the asperities and thus the adhesion-contact forces. Indeed, once an impact occurs, the energy

*E*

_{I}is converted into elastic and plastic deformations energies. The asperities loading process finishes once all the energy has been converted. The energy for elastic wave propagation is neglected in this work; however the elastic energy in the Ru film is accounted for. With these assumptions, the distance

*d*

_{e}between the two rough surfaces mean planes of asperity heights reached at the end of the impact process is deduced from

Once the distance *d*_{e} has been computed, the deformed profile of the asperities is known, and the unloading process can be studied. In particular, the adhesive contact forces *F*_{nT}^{U} are evaluated from (11.16) in terms of the distance *d* > *d*_{e}.

_{max}(

*h*) of the maximal interference reached for an asperity of initial height

*h*. From this function the profile change of an asperity of initial height

*h*can be known to evaluate its effect on the loading/unloading forces (11.16). Thus, the reliability of the micro-switch can be studied by considering the effect of repeated interactions between the movable/substrate electrodes. Indeed, the unloading curves change after repeated interactions until reaching accommodation, as illustrated on Fig. 11.4, where the unloading curves after 1, 2, 3 and 10 cycles are reported. From this figure it appears that the pull-out force after accommodation can be predicted, opening the way to stiction-free design. On this figure the elastic solution is also reported, and is shown to underestimate the pull-out force. Also the loading curve is represented.

## 11.4 Conclusions

In order to predict stiction in MEMS structures, a possible approach is to consider a multi-scale framework. If at the higher scale a finite element analysis can be considered, it requires an adhesive-contact law to be integrated on the interacting surfaces.

The definition of this adhesive-contact law constitutes the micro-scale problem. In this paper, this adhesive contact-distance curve of two interacting elasto-plastic rough surfaces was established using a semi-analytical analysis. First the deformed profile of the asperity is evaluated from literature models, which uncouple the plastic deformation from the adhesive effect. This assumption usually holds except for materials suffering from jump-in induced plasticity, as for gold, for which the sole adhesion effect can lead to plastic deformations. Then, we use Maugis-Kim adhesive theory to evaluate the adhesive-contact forces. In order to account for the deformed shape of the asperity, assumed as spherical in the Hertz contact of the Maugis theory, we propose to evaluate an effective asperity radius which depends on the interference. With this method, we can predict the loading/unloading hysteresis curves of a single elastic–plastic asperity interacting with a rigid plane. Finally a statistical model of asperity height is considered to study the interaction of two elasto-plastic rough surfaces.

The predictions of this model are illustrated by considering the cyclic loading of a 1D micro-switch application. It is shown that the repeated loading of a MEMS switch changes the structure of the contacting surface due to the plastic deformations. Thus, with time, the contact surfaces become smoother, increasing the adhesion effect. This effect should be considered at the design stage to avoid in-use stiction.

### References

- 1.Van Spengen W, Puers R, DeWolf I (2003) On the physics of stiction and its impact on the reliability of microstructures. J Adhes Sci Technol 17(4):563–582CrossRefGoogle Scholar
- 2.Do C, Hill M, Lishchynska M, Cychowski M, Delaney K (2011) Modeling, simulation and validation of the dynamic performance of a single-pole single-throw RF-MEMS contact switch. In: 2011 12th international conference on thermal, mechanical and multi-physics simulation, and experiments in microelectronics and microsystems (EuroSimE), Linz, Austria, April 2011, pp 1–6Google Scholar
- 3.Wu L, Noels L, Rochus V, Pustan M, Golinval J-C (2011) A micro-macroapproach to predict stiction due to surface contact in microelectromechanical systems. J Microelectromech Syst 20(4):976–990CrossRefGoogle Scholar
- 4.Wu L, Rochus V, Noels L, Golinval J-C (2009) Influence of adhesive rough surface contact on microswitches. J Appl Phys 106(11):113502-1–113502-10Google Scholar
- 5.Johnson K, Kendall K, Roberts A (1971) Surface energy and the contact of elastic solids. Proc R Soc Lond A Math Phys Sci 324(1558):301–313CrossRefGoogle Scholar
- 6.Derjaguin B, Muller V, Toporov Y (1975) Effect of contact deformation on the adhesion of elastic solids. J Colloid Interface Sci 53(2):314–326CrossRefGoogle Scholar
- 7.Maugis D (1992) Adhesion of spheres: the JKRDMT transition using a Dugdale model. J Colloid Interface Sci 150(1):243–269CrossRefGoogle Scholar
- 8.Kim K, McMeeking R, Johnson K (1998) Adhesion, slip, cohesive zones and energy fluxes for elastic spheres in contact. J Mech Phys Solids 46(2):243–266MathSciNetMATHCrossRefGoogle Scholar
- 9.Greenwood J, Williamson J (1966) Contact of nominally flat surfaces. Proc R Soc Lond A Math Phys Eng Sci 295(1442):300–319CrossRefGoogle Scholar
- 10.Greenwood J, Tripp J (1971) The contact of two nominally flat rough surfaces. Proc Inst Mech Eng 1847–1996 185(1970):625–633CrossRefGoogle Scholar
- 11.Jones R (2004) Models for contact loading and unloading of a rough surface. Int J Eng Sci 42(17–18):1931–1947MATHCrossRefGoogle Scholar
- 12.Williams J (2005) The influence of repeated loading, residual stresses and shakedown on the behaviour of tribological contacts. Tribol Int 38(9):786–797CrossRefGoogle Scholar
- 13.Majumder S, McGruer N, Adams G, Zavracky P, Morrison R, Krim J (2001) Study of contacts in an electrostatically actuated microswitch. Sens Actuat A Phys 93(1):19–26CrossRefGoogle Scholar
- 14.Wu L, Golinval J-C, Noels L. A micro model for elasto-plastic adhesive-contact in micro-switches. Tribol Int (submitted)Google Scholar
- 15.Chang W, Etsion I, Bogy D (1987) An elasticplastic model for the contact of rough surfaces. J Tribol 109(2):257–263CrossRefGoogle Scholar
- 16.Jackson R, Green I (2005) A finite element study of elasto-plastic hemispherical contact against a rigid flat. J Tribol 127(2):343–354CrossRefGoogle Scholar
- 17.Etsion I, Kligerman Y, Kadin Y (2005) Unloading of an elastic–plastic loaded spherical contact. Int J Solids Struct 42(13):3716–3729MATHCrossRefGoogle Scholar
- 18.Kadin Y, Kligerman Y, Etsion I (2007) Cyclic loading of an elasticplastic adhesive spherical microcontact. J Appl Phys 104(7):073522-1–073522-8Google Scholar
- 19.Du Y, Chen L, McGruer N, Adams G, Etsion I (2007) A finite element model of loading and unloading of an asperity contact with adhesion and plasticity. J Colloid Interface Sci 312(2):522–528CrossRefGoogle Scholar