Stiction Failure in Microswitches Due to Elasto-Plastic Adhesive Contacts
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.
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 . 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  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  based on the elastic micro-model  has been shown to predict accurate results for elastic materials in dry environment , 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 , until plastic accommodation or shakedown . This induces a “contact hardening”  and the pull-out force increases until accommodation, unless in-use stiction happens first.
To account for the elasto-plastic behavior, the authors  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 , we can consider the Maugis theory  completed by Kim expansion  to evaluate the adhesion forces during the unloading phase  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 . In this model only van der Waals forces are considered, which is a realistic assumption below 30% humidity . 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
11.2.2 Single Asperity Elasto-Plastic Adhesive Contacts
Because of the elasto-plastic behavior happening during contacts, the theory developed here results in different adhesive-contact forces during loading FnL(δ) and unloading FnU(δ). During the loading phase, once δCP is reached, the maximum interference is identical to the current one (δmax = δ) and the deformed profile can be evaluated from (11.4) and (11.5). Thus, the loading force FnL(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 − δres, where δres increases during the whole loading process. During unloading however, the residual (δres) and maximal (δmax) interferences reached remain constant. The adhesive-contact force during unloading FnU(δ) is computed from the Kim extension  of Maugis theory , with as input for R the effective radius (11.12), and as input for δ the effective value δ − δres. 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 .
Properties of Ru films
11.2.3 Rough Surfaces Interaction
11.3 Cyclic Loading of a Micro-switch
Properties of the micro-switch
εd/ ε0 [−]
Once the distance de has been computed, the deformed profile of the asperities is known, and the unloading process can be studied. In particular, the adhesive contact forces FnTU are evaluated from (11.16) in terms of the distance d > de.
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.
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