## Abstract

It is well known that compliant freestanding microstructures often stick to the underlying substrate due to capillary pull during wet processing. Hence, it is generally believed that dry processing, which involves only gases, avoids stiction. Contrary to this expectation, here we show experimentally that stiction may also occur during dry processing. We investigate both experimentally and theoretically possible origins of the force that brings the microstructures into contact leading to stiction in a dry environment. The study suggests that aerodynamic drag is the primary force that is responsible for dry stiction. It comes into play during venting or purging with nitrogen the vacuum chambers (reactive ion etcher (RIE)) with compliant microstructures. During the venting process, gases rush into the chamber through an inlet setting up local flows inside the chamber, and subjecting the freestanding microstructures to aerodynamic drag. If the structures are close to one another, such as two parallel structural beams anchored at the ends (studied in detail in this paper), the beam facing the flow shields the downstream beam from the flow. Thus, the upstream beam experiences a larger drag compared to the shielded one, and the gap between them reduces. Smaller is the gap, higher is the shielding effect. The upstream beam may contact the downstream one, and depending on the surface properties of the beams (such as pure aluminium surface prior to oxidation), they may stick to one another through a zipping action. Dry stiction may thus be avoided by slowing down the venting or purging process of RIE chambers with compliant microstructures.

## 1. Introduction

Stiction has been a major challenge in MEMS (micro electro mechanical systems) fabrication since its inception. It is well known that during liquid processing of released soft microstructures, attractive capillary force may develop between hydrophilic surfaces which brings them together into contact followed by stiction (Mastrangelo & Hsu 1993). Considerable effort has been directed to reduce the capillary force by surface modifications, such as using self-assembled monolayers to render the surfaces hydrophobic (Srinivasan *et al*. 1998), or by reducing the surface tension of water using CO_{2} drying (Resnick & Clews 2001). It is, generally, believed that if the MEMS processing does not involve liquids, i.e. if only gases are used for processing (dry processing), then there will be no force involved in bringing the surfaces together, and thus no stiction will occur.

We report that, contrary to the general expectation, stiction may also occur in dry processing of soft microstructures. This paper investigates and reveals the origin of the force that initiates the stiction.

The study is motivated by an apparent paradoxical observation. We fabricated a set of aluminium beams, 100 nm thick, 1 mm long and 10 μm deep. The beams were anchored at the ends, and were separated from one another by various distances. We found that the beams that were 2 μm apart were all stuck together after release from their silicon substrate. The beams were under tensile residual stress, and were electrically connected to each other. Having no liquid in between the beams and the gap between them being too large (2 μm) for van der Waals force to be effective in bringing them together, and being electrically connected so that Coulomb force, if any, can only be repulsive, the existence of an attractive force appeared paradoxical. In order to resolve the paradox, we carried out several experiments and numerical simulations which show that the force originates primarily from the aerodynamic drag of gases (nitrogen or air) flowing over the beams in the reactive ion etching (RIE) chamber during venting. Such drag force may move soft structures and bring neighbouring surfaces into contact and initiate the stiction process. Understanding this ‘dry stiction’ phenomenon will provide design guidelines for soft MEMS structures, and clues to control the venting step of the RIE chamber.

In the following, we describe the experimental observation that motivated the study and explore possible candidates for the force that may bring the beams into contact. Two possible candidates are studied in detail experimentally and theoretically. The other candidates are discarded by first principle arguments.

## 2. Problem statement

Figure 1 shows the aluminium beams with their dimensions. All such pairs were found stuck together after removing from the RIE chamber. The fabrication process flow of the beams is also shown. Aluminium was sputter deposited on silicon walls with various thicknesses. The deposited aluminium formed the top and the sidewall coatings of the walls. Aluminium was removed from horizontal surfaces by ion milling. The silicon wall was then removed by dry etching to free the coatings to form thin metal beams. Thus, fresh aluminium (without aluminium oxide) surfaces were exposed, which, when brought in contact, bonded to each other, possibly before aluminium oxide could have been formed on the mating surfaces. The aluminium beams that were not stuck due to larger gaps between neighbours appeared taut, and none of them were buckled which implies that they were possibly under residual tension.

### (a) Possible mechanisms of contact

Figure 1 shows that the aluminium beams were released at the last step of fabrication followed by venting (to atmosphere or by nitrogen for purging) of the RIE chamber. Hence, any mechanism that brings the beams together must have acted after release and during venting. Since there is no liquid filling the gap between the beams, capillarity cannot be a mechanism. Consider the following mechanisms.

*Buckling*. During isotropic Si etch, the pressure in the chamber is kept uniform, and hence possible mechanical forces due to pressure fluctuation or vibration are minimal. Hence, if the beams come into contact during this phase, they may do so by buckling due to compressive residual stress, if any. By buckling, the beams may approach each other near the mid length and contact. If so, then the beams not in contact would reduce the gap at the middle by at least 1 μm, an easily measurable quantity. Furthermore, in the absence of imperfection, each beam, if buckled, would have at least two equally probable states, i.e. it may buckle towards the neighbour or away from it. Hence, for a given gap between the beams, some pairs would contact, and some not, contrary to our observation that all pairs contact when the gap is 2 μm. Imperfections in the beams will perturb the symmetry. Imperfections may be introduced by the fabrication process that involved sputtering of Al onto a Si beam, followed by Si etching. This may lead to stresses in the Al film with a gradient along the thickness. Such stress in the two facing beams would be mirror-symmetric, making them both either buckle out or in, again changing the gap between the beams. No such gap change was observed, rather the beams appeared taut implying that they might be under tensile residual stress. Thus, buckling is discarded as a possible mechanism.*Thermal vibration*. Soft structures may vibrate due to thermal energy. The mean square deflection at the midpoint of a beam anchored at both ends, and oscillating at its first mode, can be estimated from (1/2)*k*〈*x*^{2}〉=(1/2)*KT*(Butt & Jaschke 1995), where*k*is the spring constant of the first mode,*K*is the Boltzmann constant and*T*is the absolute temperature. It will be shown later that, assuming a cosine shape for the first mode of the aluminium beam,*k*=1.135×10^{−5}N m^{−1}. Thus, at room temperature, the amplitude of thermal fluctuation is about 2 nm, much less than the minimum of 1 μm required for contact. Thus, thermal vibration is discarded as a possible mechanism.*Electrostatic force*. If the beams trap charges during plasma etching, then one expects Coulomb force to develop between the beams. However, the beams are electrically connected to each other (figure 1), and hence Coulomb force can only be repulsive as the charges will be distributed over both the beams. Thus, electrostatic force is discarded as a candidate mechanism for contact.*Forces during venting—squeeze film effect*. At the beginning of the venting process, air rushes into the RIE chamber as the pressure increases. Air also rushes into the gap between the beams to equilibrate the pressures in the gap and the ambient. If there is a time delay in equating the two pressures, then momentarily the outside pressure will be higher than that within the gap, thus generating a net attractive force between the beams (figure 2*a*,*b*). The motion of the beam will be determined by the force history and the beam's dynamic characteristics. This mechanism of forcing is similar to squeeze film damping (Langlois 1962; Veijola & Mattila 2001) where two parallel plates in air with a small gap between them approach towards or recede from each other at high speeds, and the pressures in the gap and in the ambient differ due to the time required for air to move out or rush into the gap. For convenience of presentation, we name this mechanism the squeeze film effect, and it will be studied in detail in §3, both experimentally and theoretically.*Air drag*. As the RIE chamber is vented, air flows into the chamber and the wafer with the released beams is subjected to the flow. Now, if the flow is transverse to the beams, then the beam facing the flow will experience a drag force. If the gap between this beam and its neighbour (downstream) is small, then the latter, being shielded by the former, will experience smaller drag. The first beam will approach its neighbour and may contact it. If, on the other hand, the gap is large, the shielding effect is minimal and both the beams will move together thus avoiding contact. This mechanism will also be studied in detail in §3, and will be shown to be the primary mechanism for dry stiction.

## 3. Investigation of two possible forcing mechanisms

In the following we study in detail, both theoretically and experimentally, the forces during venting due to the squeeze film effect and drag.

### (a) Squeeze film effect

#### (i) Analysis

We first carry out a two-dimensional fluid dynamics analysis of a pair of beams, 12 μm from one another and 20 μm tall. These dimensions are chosen to be consistent with the geometry of the test beams discussed in the experimental section. The gap between the beams is initially at vacuum. The outside pressure is atmospheric. Air begins to fill the gap from time *t*=0. The gap pressure thus increases with time, while the outside pressure is kept constant. Our primary interest is in the determination of the characteristic time *τ* necessary to fill the gap for two reasons: (i) to extrapolate the characteristic time for any other cavity, particularly the *RIE chamber*, and (ii) to estimate the forcing history on parallel beams due to the squeeze film effect *alone* in the RIE chamber. The computational fluid dynamics code, Fluent 6.2.1, is employed for the analysis. Figure 2*c* shows the simplified model used in the analysis. Since the beams are long and flow is expected to be similar at any section of the beams (except near the ends), we treat the problem as two-dimensional. To further simplify the problem, we neglect the effect of silicon substrate 20 μm below the lower edge of the beams. Thus, air flows to fill the gap through both the top and the bottom openings, and the problem is symmetric with respect to a horizontal line at the mid height of the beams. The beams are assumed rigid for the fluid dynamics analysis, i.e. they do not move due to the pressure differential. The assumption is based on the expectation that the time to fill the gap is much smaller than the period of first resonance of the beams (to be verified). Hence, the beams move by only a small fraction of the initial gap when the force has already decayed to near zero. The dynamics of the beams in response to the forcing history is evaluated independently of the air flow problem, i.e. the two problems are decoupled. Room temperature is prescribed as the initial condition for the entire domain (gap and outside). The boundary conditions are: no flow through the beams, no slip is allowed along their walls, and the pressure outside the gap remains at atmospheric. Air is treated as compressible, and a continuum analysis is assumed to hold even at the microscale. Fluent evaluates pressure, density, temperature and velocity fields of air in the domain of the gap with increasing time. Force per unit length of the beam is obtained from the pressure distributions on both faces of the beams. Note that the pressure outside the gap is constant, but its distribution on the inner wall (bounding the gap) changes with time. The net force acts along the transverse direction of the beam. The net shear force is zero due to symmetry. The properties of air used in the analysis are: density *ρ*=1.2 kg m^{−3}, dynamic viscosity *μ*=1.82×10^{−5} N s m^{−2} at 20 °C and at standard atmospheric pressure.

Figure 3 shows the time evolution of the average pressure on the inner surface (bounding the gap) of a beam, normalized with respect to the ambient atmospheric pressure. The result also represents the net force on the inner surface normalized by the constant net force on the outside surface due to atmospheric pressure. Note that the gap pressure increases to two times the outside pressure. It then oscillates about the outside pressure with decreasing amplitude. This overshoot of pressure is possibly because a small gap is being filled by two large inlets. Due to the overshoot, the beams will repel each other (after initial attraction) until the pressures equilibrate.

##### 3.1.1.1 A simple model

In order to interpret the above result, and to determine the characteristic time of filling the gap, we construct a simple model. Let *P*(*t*) be the average pressure at time *t* within the gap, and *P*_{0} be the ambient pressure. Then the mass flow rate, d*m*/d*t*, into the gap is(4.1)where *h* is a constant proportional to the cross-sectional area of inlet. Since, the temperature of the RIE chambers are controlled by cooling systems, we do not expect the temperature of the chamber (metallic) as well as the air filling the chamber to rise significantly during the process of venting. Thus, assuming constant temperature, *T*, and ideal gas law, *PV*=*nRT*, where *V* is the volume of the gap, *R* is the universal gas constant, *n* is the number of molecules in the fixed volume *V*, we have(4.2)where *C*=constant∝1/*V* and *m* is the mass of the gas in *V*. From equations (4.1) and (4.2)(4.3)where(4.4)With *P*(0)=*P*_{i}, the solution for *P* is(4.5)If *P*_{i}=0(4.6)Thus, *τ* represents the time required for the pressure to increase to (1−e^{−1}) of its final steady value. As *t*→0, (d/d*t*)(*P*/*P*_{0})→1/*τ*. From the fluid dynamics analysis (figure 3), the initial rate of change of *P*/*P*_{0} is 1/0.059×10^{−6} s^{−1}. Then *τ*=16.95 μs for the gap. Pressures predicted by equation (4.6) with *τ*=16.95 μs is also shown in figure 3, which does not show any overshoot, as expected, but shows good match with the fluid dynamics analysis for the time to equilibrate the gap and the outside pressure.

In the above analysis, *P*_{0} is considered constant. In reality, *P*_{0} is the pressure in the chamber (RIE chamber) at the location of the beams which increases with time as the chamber is vented. In order to estimate the time dependence of *P*_{0}, we model the venting process similar to the process of filling the gap between the beams. We assume ideal gas law, constant temperature, and that the inlet valve does not choke during venting. Then the time evolution of the average pressure within the chamber can be shown to be(4.7)where *f* is the constant atmospheric pressure outside the chamber, and *β* is the, as yet unknown, characteristic time of filling the chamber. Here, the initial pressure of the chamber is considered zero. In order to get an estimate of *β*, we note from equation (4.4) that the characteristic time is proportional to the volume of the chamber and inversely proportional to the inlet area. Since the characteristic time, *τ*, for the gap between two beams is already known, we have(4.8)where *V* is the space being filled, and *A* is the inlet area. The volume of the gap between the two beams (per unit length normal to paper), *V*_{two beam}=12×20 μm×1 m and the total inlet area, *A*_{two beam}=2×12 μm×1 m. The geometry of RIE chambers vary widely, but a typical size would be 20 cm in radius and 20 cm tall. If the inlet has a diameter of 2 cm, then, from equation (4.8), *β*≈100 s, a typical venting time for most RIE chambers. The characteristic time for the aluminium beams with 2 μm wide and 10 μm tall gap can similarly be found to be *τ*_{Al beam}=16.95/2=8.47 μs.

With *P*_{0} as a function of time given by equation (4.7) the pressure, *P*(*t*), within the gap between the beams in the chamber is governed by(4.9)Then(4.10)Corresponding pressure differential(4.11)If *β*≫*τ*, it can be shown that the pressure differential reaches a maximum value of(4.12)For *β*=100 s, *τ*_{Al beam}=8.47 μs, *f*=101 325 Pa, equation (4.12) gives (*P*_{0}−*P*)_{max}=0.0085 Pa at *t*=1.4×10^{−4} s. Note that this maximum pressure differential will generate a force of 0.085×10^{−6} N m^{−1} on the aluminium beams (10 μm deep). The dynamic response to this force is evaluated next.

##### 3.1.1.2 Dynamics

In order to determine the motion of the beams due to a uniformly distributed loading, we approximate the deformed shape (first mode shape) of the beam by , where *z* is the mid displacement and *x* is the position along the beam from the support. The shape closely matches the deflected shape of a beam anchored at both ends and subjected to a uniformly distributed static load. We will use the shape *ψ*(*x*) to determine *z* for both static and dynamic loads. Since, the beams under consideration are long and slender, the strain energy of deformation will be contributed by both bending and stretching, giving rise to a linear and cubic terms in the force–displacement relation. The lumped parameters of the beam are (Clough & Penzien 1986)(4.13)Here *E* is elastic modulus, *K*_{1} and *K*_{2} are the linear and cubic spring constants, *M* and *F* are the equivalent mass and force, *A*, *I*, , and *w* are the cross-sectional area, moment of inertia, mass and force per unit length of the beam, respectively. When *w* is static (independent of time), the mid displacement *z* is given by(4.14)When *w* is time dependent, *z* is governed by the nonlinear differential equation(4.15)A damping term may be added to equation (4.15) if needed. For pressure differential, *P*_{0}−*P*, we have *w*=*H*(*P*_{0}−*P*), where *H* is the height of the beam.

For the aluminium beams, 0.1 μm thick, 10 μm tall and 1 mm long, *K*_{1}=1.136×10^{−5} μN μm^{−1}, *K*_{2}=8.54×10^{−4} μN μm^{−3}, *M*=1.05×10^{−12} kg. When the forcing is due to *P*_{0}−*P* (equation (4.11)), and *β*=100 s, *τ*=*τ*_{Al beam}=8.47 μs, *f*=101 325 Pa (atmospheric pressure), then , from equation (4.15), represents the deformation of the beam while the RIE chamber is vented. Figure 4 shows the variation of *z*(*t*) as a function of time. Note that the maximum displacement is about 0.6 μm, less than 1.0 μm necessary for contact between two beams 2 μm apart. Note that the maximum displacement is not very sensitive to the value of *β*. For example, *β*=200 and 50 s give the maximum displacement of about 0.5 and 0.75 μm, respectively.

#### (ii) Experiment

We will now test some of the theoretical predictions on squeeze film effect by observing the dynamics of two parallel beams subjected to time varying ambient pressure. The test beams are made of single crystal silicon using the SCREAM process (Shaw *et al*. 1994) with the dimensions 0.5 μm×20 μm×5 mm (figure 5*a*). The beams are aligned along the (110) crystal orientation of silicon. The small thickness of the beams is achieved by thermal oxidation of thicker beams, and subsequent etching of the oxide by hydrofluoric acid. The gap between the beams is 12 μm. It was not possible to fabricate beams with smaller gaps due to capillary driven stiction during wet processing. Note that these beams have the same thickness/length ratio as the aluminium beams of figure 1. But their compliance is less than that of the aluminium beams by the modulus ratio *E*_{Al}/*E*_{Si(110)}=70/170, although a tensile residual stress in the aluminium beams will increase their lateral stiffness. Silicon beams are expected to be free from residual stress. The lumped parameters for the beams (equation (4.13)) are *K*_{1}=5.519×10^{−5} μN μm^{−1}, *K*_{2}=1.66×10^{−4} μN μm^{−3} and *M*=43.69×10^{−12} kg. The first resonant frequency rad s^{−1}=179 Hz.

The ambient pressure for the beams is varied periodically by a sinusoidal acoustic wave which results in compression and rarefaction at any point in space. If the frequency of the wave is low (in the order of the first resonant frequency of the beams, *ca* 200 Hz), then the corresponding wavelength is large, in the order of a metre. If a pair of parallel microbeams, a few millimetres long and with a few micrometres gap between them, are placed in the acoustic environment, then the entire span of the beam will experience uniform ambient pressure at any instant of time, although the pressure will change periodically with time. Now, if there is a time delay in equilibrating the ambient and the gap pressure, and sufficient force is generated, then the beams will approach towards, and recede from, each other periodically.

Figure 5 shows the experimental set-up and the silicon beams used in the study. A speaker, powered by a function generator, produces the sinusoidal acoustic wave. A microphone measures the acoustic signal at any point away from the speaker and displays the result through an oscilloscope. A pair of microbeams is placed in front of the speaker, and their motion is observed by a high-speed camera (Photron FASTCAM APX High Speed Imaging System) with pictures taken at 50 000 frames per second.

Three experiments are carried out.

The speaker is placed 38 mm away from the beams. The intensity of the acoustic wave is adjusted so that the motion of the beams can be measured with confidence (amplitude of motion at mid length is much larger than the resolution of measurement) when they are excited at the resonant frequency. Figure 6 shows the motion of the beams at resonance (around 250 Hz). The beams indeed move a large distance, but contrary to the expectation that the motion will be symmetric, i.e. the beams will approach towards or recede from each other, they both move along the same direction. Thus the motion is not induced by the pressure differential between the ambient and the gap. We also note that the beam facing the incident wave has much larger amplitude of motion than the downstream beam. This unexpected observation raised the possibility that the beams are responding to drag force that might be generated by the flow of air due to the motion of the diaphragm of the speakers. Thus, the downstream beam moves less than the upstream one, being shielded by the latter, but both move along the same direction.

In order to eliminate the effect of drag (which may be introduced by the proximity of the speaker to the beams), we place the speaker further away from the beams (203 mm instead of 38 mm), and place a foam filter in front of the speaker to damp out any air flow. The acoustic wave intensity at the source (speaker) is increased so that the incident wave intensity at the location of the beams is similar to that in the first experiment (measured by the microphone) and the beams are subjected to primarily an ambient change of acoustic pressure with minimal air flow. Now, no perceptible motion is observed in the beams subject to the acoustic pressure waves.

The silicon chip with the beams is now rotated by 90° and the speaker is placed 38 mm away from the end of the beam without any filter. Hence, even if there is air flow, no drag force is generated perpendicular to the beams. The beams are subjected to almost a uniform ambient pressure (since the length, 5 mm, is small compared to the acoustic wavelength of about a metre) which changes with time. Again, we observe no perceptible motion of the beams with, or without, a foam filter in between the speaker and the beams.

In order to explore whether the squeeze film effect can be induced at high frequency, we sweep the frequency of the acoustic waves in the range 0–10 000 Hz (the gap between the beams is still much smaller than the acoustic wavelength) in each of the above experiments. No squeeze film effect was observed.

In order to interpret the experimental observation, we analyse the dynamics of the beam due to acoustic excitation. The pressure differential, *P*_{0}−*P*, between the gap and the ambient is obtained from the steady-state solution for *P*(*t*) from equation (4.3) with *P*_{0}=*f* sin(*ωt*). Thus(4.16)Here, *f* is the amplitude of the acoustic wave (*ca* 30 μPa) and *ω* is the acoustic frequency (rad s^{−1}). Since, in the acoustic experiment, *ω* was chosen at the resonant frequency of the beams, we treat the dynamics of the beam with small damping (*ca* 1% of critical damping). Then, the mid deflection, *z*, satisfies(4.17)where *ξ*=0.01 is the first modal damping coefficient and . Solution of equation (4.17), when *ω*=*ω*_{0} shows the amplitude of *z*, is in the order of 0.025 μm. It is thus not a surprise that experimentally the motion could not be detected using optical microscopy. Clearly, the motion of about 2 μm observed during experiment (1) is not due to *P*_{0}−*P*, but due to aerodynamic drag. In the following, we study the effect of drag in detail, first experimentally and then theoretically.

### (b) Air drag

#### (i) Experiment

The schematic of the experiment is shown in figure 7. Here, the silicon chip carrying a set of parallel beams is placed in a uniform air-flow field in a wind tunnel such that the flow is normal to the beams. Similar beams were used to study the squeeze film effect. Since the air flows over the silicon substrate before meeting the beams, a boundary layer develops, and the beams are embedded in the layer. In order to minimize the edge effect on the boundary layer due to the edge of the silicon chip, we place the chip on a thin aluminium plate that extends by 20 cm from the edge of the chip along the upstream direction of flow. Thus, the beams are far from the edge.

The experimental set-up consists of a wind tunnel (figure 7). Compressed air is fed through the end of the tunnel through a flow control valve. Thus, the flow velocity can be controlled. An anemometer with a 4 μm diameter wire (1.2 mm long) is used to measure the flow velocity. For all the flow conditions used in this study, we observed uniform velocity field at the mid region of the wind tunnel cross-section. The uniformity spans a region much larger than the size of the silicon chip. The uniformity is maintained along the entire length of the tunnel, except near the ends. We will denote this uniform velocity as free stream velocity, *U*. By increasing the flow, we simply increase the free stream velocity which is measured to define the flow condition. The silicon chip on the aluminium plate is placed at mid height of the tunnel by a small pedestal. The top cover of the tunnel is made of glass for visualization of the beams using a microscope and a camera.

The experiment involves two tasks. First we obtain the velocity profile in the boundary layer at a location 2.5 mm upstream from the beam. For a given *U*, we measure the flow velocities at this location using the anemometer at different heights from the silicon substrate (figure 8). Due to the size restriction of the anemometer, the velocities could not be measured within 50 μm from the substrate. Hence, the profile is extrapolated to the substrate. We employ Blasius boundary layer solution (Munson *et al*. 1998) for flow over a flat plate for the extrapolation. Blasius solution predicts that near the plate surface, flow velocity, *u*, varies almost linearly with the distance *y* from the surface as(4.18)where *x* is the distance from the edge of the plate, *ν* is the kinematic viscosity and *δ*(*x*) is the thickness of the boundary layer defined such that at *y*=*δ*(*x*), *u*=0.99*U*. From a linear fit (figure 8) of the velocity profile near the surface, we obtain *δ*=5.65 mm. Thus, the extrapolated velocity distribution is given by(4.19)This velocity profile will be used in flow simulation around the beams. The profile will be prescribed at 200 μm upstream from the beam facing the flow. Since, the velocity profile is measured at nearly *x*=200 mm away from the edge (2.5 mm away from beam), the change of *δ* due to increase in *x* by 2.3 mm is negligible, and hence is ignored.

In the second task of the experiment, we measure the deformation of the beams at a given flow condition. Note that this is the steady-state deformation due to air drag. Figure 9 shows the mid displacement of the pair of beams as a function of free stream velocity. Note that the first beam (facing the flow) deforms more than its neighbour, as expected, since the first one shields the second from the flow. Using the same set-up, we also measure the deformation of a single beam (instead of a pair) at different free stream velocities (figure 9) to explore whether the gap between the pairs introduces any unexpected forcing mechanism at microscale. It turns out that the two deformations are comparable.

In order to interpret the experimental findings, we estimate the drag force on the pair of beams by fluid dynamics simulation of the flow using Fluent 6.2.1. The corresponding beam deformations are evaluated using elasticity theory (equation (4.14)) and are compared with experimental results.

#### (ii) Analysis

The analysis is carried out for one data point in figure 9 where the free stream velocity *U*=1.44 m s^{−1}. The corresponding boundary layer velocity profile is shown in figure 8. Figure 10*a* shows the model used for the analysis of fluid flow. Here the flow is assumed two-dimensional, incompressible and at steady state. The beams are assumed rigid so that they do not move due to flow. The deflection is calculated after the force is evaluated, so that the fluid and the elasticity problems are treated as decoupled. In reality the flow will deflect the beams and change the gap between them from the initial value of 12 μm to around 10 μm (experimentally observed, and to be shown later by analysis). The change in gap will induce a change in flow pattern and, hence, the drag on the beams. Thus, one needs to solve the coupled problem iteratively. We avoid this iterative process both for simplicity and since most of the relevant physics can be captured with the uncoupled analysis.

Figure 10*b* shows the flow velocities around a pair of beams 12 μm apart. Clearly the upstream beam shields the downstream beam, and the drag force on the second beam is less than on the first. From the pressure distribution on the walls of the beam, the net horizontal component of force (per unit length) on the first and the second beams are found to be 4.09×10^{−6} and 2.3×10^{−6} N m^{−1}, respectively. Equation (4.14) predicts the corresponding deflections due to the above forces as 3.92 and 3.22 μm. The experimental deflections are 1.75 and 1 μm, respectively, close to the predicted values, considering the simplifications made in arriving at the predictions.

From the above theoretical and experimental studies on squeeze film and air drag, we draw the following conclusion. As the RIE chamber is vented, air flows into the chamber and begins to raise its pressure. In the process, it generates two force histories on the parallel beams, one due to flow of air into the chamber that creates a drag on the beams, the other due to squeeze film effect. The former is much larger than the latter. If the beams are close to each other, then the beam facing the flow shields its neighbour from the flow and experiences a relatively larger drag. Thus, it approaches the neighbour and forms a contact leading to dry stiction. The smaller the gap, the larger is the shielding effect and the higher is the probability of contact. Note that the smaller the gap, the higher is the differential pressure as well, and the forcing histories due to both the differential pressure and the drag contribute to bring the beams together.

## 4. Discussion

We set out to resolve the question: what force brings the two aluminium beams into contact, i.e. what may initiate dry stiction of soft microstructures close to each other. Considering several possibilities, we concluded that the beams of figure 1, 2 μm apart from one another, were brought into contact by the drag force as the RIE chamber was vented. However, until now, we do not have an estimate of the flow profile or the drag-induced force on the beams inside the chamber. The force may be estimated by a complete three-dimensional analysis of the flow within the chamber and its valve system as it is vented. However, we take an alternative route to get an order of magnitude estimate of the flow from a simple experiment.

It is well known that 3–4 in. silicon wafers on the platen of vacuum chambers (RIE, deposition chambers) are often found to have moved from their original location during venting. Some vacuum chambers allow slow and controlled venting to avoid this problem. Clearly, flow of air within the chamber during venting causes enough force on the wafer to move it against frictional resistance. Thus, in order to get an estimate on the force, we measure the frictional coefficient between a 4 in. wafer and a platen (PlasmaLab Freon/O2 Reactive Ion Etcher (RIE) System) by tilting the platen with the wafer until the wafer slides due to its weight. From the angle and the weight, we get the friction force as 0.02 N, and the friction coefficient as 0.2.

In order to obtain an estimate on the air flow in the RIE chamber during venting, we assume that the flow induced shear traction on the wafer is just enough to overcome the friction, and that any other force on the wafer due to flow, such as lift or the force on the small vertical surface (0.5 mm tall, thickness of the wafer) of the wafer on its edge, is negligible. Furthermore, we assume that the wafer is placed in a free stream flow with a velocity *U*. The flow over the wafer creates a boundary layer with a shear stress (Munson *et al*. 1998), , where *ρ* is mass density, *μ* is dynamic viscosity, and *x* is the distance from the leading edge of the wafer. Replacing the circular wafer of radius *R* by a square plate with width 2*R*, the total force, *F*_{D}, on the plate can be obtained by an integration. Thus, *F*_{D} overestimates the force on the wafer. Equating *F*_{D} with the experimentally measured friction force of 0.02 N, we get(4.20)For *R*=2 in.=5.08 cm and *ρ*=1.2 kg m^{−3}, *μ*=1.82×10^{−5} N s m^{−2} (at atmospheric pressure and at 20 °C), we get *U*=34 m s^{−1}, much larger than *U*=1.44 m s^{−1} in the drag experiment that generated a force of about 4.09×10^{−6} N m^{−1} on the beam (predicted by Fluent, and verified by experimental observation of beam deflection).

A fluid dynamic analysis with two parallel beams, 20 μm tall, 2 μm apart from one another, and subjected to *U*=1.5 m s^{−1}, shows a force of 2.77×10^{−5} N m^{−1} on the upstream beam, and 7.38×10^{−6} N m^{−1} on the downstream beam acting in the reverse direction. If the beams were made of aluminium and were 0.1 μm thick, then the upstream beam would deform by *z*=2 μm (equation (4.14)) along the direction of flow, whereas *z*=2 μm for the downstream beam against the flow. The beams thus would contact each other. It is not surprising that all the aluminium beams were found stuck to each other due to a possible free stream velocity of *U*=34 m s^{−1}.

To summarize the rationale of the steps taken to investigate drag-induced stiction, we first measured the deflection of a single and a pair of silicon beams, 12 μm apart from each other, subject to air flow in a wind tunnel. The experiment is then simulated by fluid dynamics code and elasticity analysis to predict the deflections. The correspondence between theory and experiment justified further analysis with the beams made of aluminium and 2 μm apart from one another. The analysis predicts contact between the beams when the free stream velocity is 1.5 m s^{−1}. To relate the analysis with the dry stiction observed in aluminium beams in the RIE chamber (figure 1), we estimated the free stream flow velocity in the chamber during venting based on the well known observation that wafers in RIE chambers occasionally move on the platen during the venting process. The estimated free stream velocity, 34 m s^{−1}, much larger than the 1.5 m s^{−1}, supports the hypothesis that the aluminium beams were brought in contact during the venting process by air drag.

## 5. Conclusions

It is, generally, understood that dry processing of compliant microstructures avoids stiction in contrast to wet processing where they are brought into contact by capillary force during drying. Here we report that stiction may occur during dry processing as well. We demonstrate dry stiction in pairs of parallel aluminium beams. The beams are 2 μm apart from one another, 100 nm thick, 10 μm tall, and 1 mm long. They were released in an RIE chamber from silicon substrate by dry processing (*SF*_{6} release), and the chamber was then vented to atmospheric pressure.We investigated the origin of the force that brought the beams into contact leading to stiction. We considered several possible forcing mechanisms and identified two of them as the most probable ones that may bring the beams into contact. These forces come into play during venting or purging the RIE chamber (with the released beams) when nitrogen or air rush into the chamber resulting in: (i) increase of pressure in the chamber and (ii) local flows within the chamber. One of the forcing mechanisms originate from the time delay in balancing the pressure within the gap between the beams and the ambient pressure. Thus, a relative suction may develop within the gap which may bring the beams towards one another. The second forcing mechanism is the aerodynamic drag on the beams due to the local flows within the RIE chamber. Here, the beam facing the flow shields the downstream beam from the flow, and hence is subjected to higher drag than the downstream one. Thus it undergoes larger deformation towards the downstream beam and the their gap reduces. Consequently, the shielding effect increases. The aerodynamic force may bring the beams into contact leading to stiction.

We investigated these two forcing mechanisms by fabricating parallel silicon beams, 12 μm apart from one another, but with similar compliance as that of the aluminium beams. The silicon beams were subjected to acoustic pressure waves with a range of frequencies so that the ambient pressure around the beams can be varied at different rates. The beams were then subjected to a range of aerodynamic drag in a miniaturized wind tunnel by varying the free stream velocity. In both cases, beam deflection were measured. The forces around the beams were estimated by computational fluid dynamics analysis using Fluent 6.2.1. The dynamics of the beams was modelled with both the linear and cubic stiffness terms contributed by the flexure and stretch deformations. The experimentally measured deformations match reasonably well with the predicted values. The experiments and the analyses suggest that the aerodynamic drag plays a dominant role in reducing the gap between compliant microstructures.

In order to relate the wind tunnel study on silicon beams with the dry stiction of aluminium beams in the RIE chamber, we estimated experimentally the free stream velocity of flow (parallel to the substrate) in the RIE chamber during venting. We find that the velocity can be an order of magnitude higher than the velocities used in the wind tunnel experiments. Based on the aerodynamic force estimates from the wind tunnel, we conclude that aerodynamic drag in the chamber was responsible for bringing the aluminium beams into contact. Dry stiction can thus be avoided by reducing the rate of venting the RIE chambers with compliant microstructures.

## Acknowledgements

The project was supported by the National Science Foundation under grant number NSF ECS 0083155. The microstructures were fabricated at the Center for Nanoscale Science and Technology, University of Illinois at Urbana-Champaign.

## Footnotes

- Received April 29, 2005.
- Accepted September 12, 2005.

- © 2005 The Royal Society