Adhesion of micro-cantilevers subjected to mechanical point loading: modeling and experiments

Journal of the Mechanics and Physics of Solids 51 (2003) 1601 – 1622

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Adhesion of micro-cantilevers subjected to mechanical point loading: modeling and experiments Edward E. Jonesa , Matthew R. Begleyb;∗ , Kevin D. Murphya a Department

b Structural

of Mechanical Engineering, University of Connecticut, Storrs, CT 06269-3139, USA and Solid Mechanics Program, Department of Civil Engineering, University of Virginia, Charlottesville, VA 22904-4742, USA

Received 8 August 2002; received in revised form 13 February 2003; accepted 14 February 2003

Abstract This paper presents experimental and theoretical results that characterize the adhesion of MEMS cantilevers by means of mechanical actuation. Micro-cantilever beams are loaded at various locations along the freestanding portion of the beam using a nanoindenter. Transitions between three equilibrium con7gurations (freestanding, arc-shaped, and s-shaped beams) and the response to cyclic loading are studied experimentally. The resulting mechanical response is used to estimate the interface adhesion energy (using theoretical models), and to quantify the energy dissipated during cyclic loading. The experiments reveal interesting behaviors related to adhesion: (i) path dependence during mechanical loading of adhered beams, (ii) history dependence of interfacial adhesion energy during repeated loading, and (iii) energy dissipation during cyclic loading, which scales roughly with estimated cyclic changes in the size of the adhered regions. The experimental results are interpreted in the context of elementary fracture-based adhesion and contact models, and brie8y discussed in terms of their implications regarding the nature of adhesion and future modeling to establish adhesion mechanisms. ? 2003 Published by Elsevier Science Ltd. Keywords: Stiction; Adhesion energy; Dissipation; Indentation

1. Introduction Adhesion is a critical issue in many micro-electro-mechanical system (MEMS) applications, as it plays an important role in the fabrication process yield, device reliability, ∗

Corresponding author. Tel.: +1-434-243-8728; fax: +1-434-982-2951. E-mail address: [email protected] (M.R. Begley).

0022-5096/03/$ - see front matter ? 2003 Published by Elsevier Science Ltd. doi:10.1016/S0022-5096(03)00025-5

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L t h free-standing

(a)

arc-shaped

(b) xo F

(c)

s

Fig. 1. Three possible con7gurations of the micro-cantilever beam: (a) freestanding, (b) sticking at just the tip, or arc-shaped, and (c) an s-shaped beam where a 7nite sized portion is adhered to the substrate. The variables used to describe the point load are shown in (c).

and the operational requirements of the device. The 7rst two of these considerations are often addressed under the umbrella of “stiction-failure”, which refers to the scenario wherein the forces driving decohesion are not suDcient to separate the components (as required for normal operation). In addition to such failures, adhesion plays a signi7cant role in device operation by dictating the forces and power required to operate devices that are designed for intermittent or cyclic contact. Although adhesion occurs naturally at all scales, it is particularly troublesome in MEMS because their small dimensions, typically large aspect ratios and close proximity of adjacent surfaces. These geometric factors imply that the strain energy of a deformed structure (which often provides the driving force for decohesion) is on the same order of the interface energy associated with the adhered contact. This can be illustrated by simply comparing the strain energy and adhesion energy for a given geometry, such as the cantilever beam illustrated in Fig. 1. For a beam de8ected into an s-shape (as in Fig. 1c), the ratio of strain energy to the interface adhesion energy is given by  2 Strain energy 1 Et
t 2 h ≡ ; (1) = Interface adhesion energy 2 i s s where E is the modulus of the beam, t is the beam thickness, h is the initial gap separation, s is the length of the freestanding portion beam, and i is the energy per unit area associated with adhesion. For convenience Eq. (1) assumes the adhered length is equal to the unstuck length. For 1, strain energy dominates and adhesion is negligible; such is the case for macroscopic beams. For ∼1, however, the energy

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associated with adhesion is comparable to the strain energy and cannot be reasonably neglected. Despite the fact that Eq. (1) is not generally applicable to other geometries, it clearly illustrates the combination of factors that lead to stiction failure in devices. Long, slender devices that experience comparatively small de8ections (and hence small amounts of total strain energy) will be prone to adhesion due to well-matched competition between adhesion energy and strain energy (which drives decohesion). For a cantilever with E ∼ 200 Gpa; t ∼ 1 m; s ∼ 100 m and h ∼ 1 m, the ratio  is approximately unity for adhesion energies on the order of 1 mJ=m2 . Adhesion energies on this order can arise from a variety of mechanisms; the list of usual suspects includes micro-capillaries due to humidity, van der Waals forces and electrostatic attraction (Israelachvili, 1992). Micro-cantilevers oHer attractive advantages for studying adhesion in small structures, owing to the ease of fabrication and the ability to vary  over many orders of magnitude by varying the beam dimensions. These two factors allow one to estimate the interface adhesion energy i by identifying quantitative relationships between beam dimensions and transitions in adhered geometry (such as those shown in Fig. 1). This was pioneered by Mastrangelo and Hsu (1992), who calculated the adhesion energy from the shortest arc-shaped beam that would remain adhered to the substrate. De Boer and co-workers expanded on this approach to consider the three con7gurations shown in Fig. 1, and established the mechanics needed to infer i from fracture mechanics models for decohesion (de Boer and Michalske, 1999). In their approach, the adhesion energy is determined from the unstuck length by equating it to the energy release rate for interface decohesion, as dictated by the unstuck region which is treated as a crack. De Boer and co-workers illustrated that the interface energy inferred from arc- and s-shaped beams is diHerent, and noted that the s-shape beams are more desirable, since the apparent 1 adhesion areas are large and less in8uenced by surface irregularities. The electrostatic forces used to actuate the beam in these studies can be considered as an external distributed load, analogous to the point load considered here (Jensen et al., 2001). Moreover, they have explored the eHects of humidity and surface coatings on the inferred adhesion energy, and have provided insight regarding the nature of adhesion mechanisms (de Boer et al., 2000). de Boer et al. (2001) have illustrated that adhesion is strongly in8uenced by nanoscale gap separations where close-range attractive forces are prominent. In the present study, adhesion is explored in the context of the response of microcantilevers to mechanical probing. Mechanical loading oHers two important advantages for exploring adhesion behavior over previously applied interferometric techniques. First, the mechanical response of a structure can be related to adhered areas, without having to visualize the system. This may enable adhesion measurements for adhered geometries where interference fringes are diDcult to detect. Second, transitions between adhered states and associated energy changes can be induced mechanically, 1 The true adhered interface area (or contact area) is governed by point-contact of nanoscale asperities; further details of the relationship between true and apparent adhered areas is given in de Boer et al. (2001).

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with forces and displacements captured via the probe. The associated energy changes (including those during cyclic loading and bond/debond cycles) can be calculated directly from load–displacement measurements. Forces, displacements and energy changes during adhesion transitions and during cyclic loading have yet to be directly quanti7ed, since previous studies have used electrostatic actuation where force is not known directly. The overall objective of this paper is to establish the mechanics of an adhered micro-cantilever beam subject to point loading, and investigate several cyclic loading scenarios in the context of adhesion energy and energy dissipation. A compressive point load is applied to the beam in the unstuck region of the cantilevers (Fig. 1c); the resulting load–displacement relationship is used along with mechanics models to infer the adhesion energy. 2 The experiments shed new light on adhesion by characterizing changes during transitions in adhered states, diHerences between loading and unloading, and energy dissipated during cyclic loading. The accompanying theoretical framework can be used both to interpret experiments and to estimate other important quantities, such as the force and energy required to detach a stiction-failed device. It should be noted that this brief introduction has been limited to an overview of explicit studies on the mechanics of adhesion in micro-cantilevers. A complementary body of work exists which focuses on the eDciency of various release procedures in producing freestanding beams (e.g. Abe et al., 1995; Rogers and Phinney, 2001). Such processing studies investigate adhesion mechanisms resulting from fabrication (e.g. capillary behavior during drying, Abe et al., 1995), and typically study release procedures and processing steps designed to eliminate stiction (e.g. laser irradiation, Rogers and Phinney, 2001). In contrast, we consider the mechanical response of adhesion transitions and cyclic loading in various states, including initially freestanding beams that are re-adhered via mechanical loading. To our knowledge, the only previous work considering the adhesion of initially freestanding beams is that of de Boer and co-workers, who used electrostatic actuation to adhere micro-cantilevers and interferometry to characterize the geometry (de Boer and Michalske, 1999; de Boer et al., 2000, 2001; Jensen et al., 2001).

2. Theoretical results for point loads applied to adhered beams In this section, we outline elementary models for the mechanical response of an s-shaped adhered beam to an applied point load, such as that shown in Fig. 1c. The adhered portion of the beam is assumed to be rigidly 7xed to the substrate, with displacements equal to the gap separation, h, and with a zero slope at the edge of the unstuck length, s. The displacement and slope of the beam at the post are assumed to be zero, and the length of the adhered region is considered in7nite. Under these 2 Ideally, one would like to pull up on the beam to separate the surfaces; the far more sophisticated fabrication procedures to produce such specimens are currently being pursued.

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assumptions, elementary beam theory yields the following results for the de8ection of the beam: (0 ¡ x ¡ x0 ) = −

1 (Fs3 − 3Fsx02 + 2Fx03 + 12hEI )x3 6 s3 EI

+

1 (Fx0 s2 + Fx03 − 2Fsx02 + 6hEI )x2 s2 EI 2

(2a)

and (x0 ¡ x ¡ s) =

1 (3Fsx02 − 2Fx03 − 12hEI )x3 1 (2Fsx02 − Fx03 − 6hEI )x2 − 3 6 s EI 2 s2 EI

1 (Fx02 )x 1 Fx03 − ; (2b) 2 EI 6 EI where (x) is the downward de8ection from the initially freestanding position, F is the applied point load, s is the unstuck length, x0 is the location of the applied load, h is the gap separation, and I is the moment of inertia. In the limit that F → 0, Eq. (2) reduces to the unloaded beam result reported elsewhere (e.g. Mastrangelo and Hsu, 1992; de Boer and Michalske, 1999; Rogers et al., 2002). A fracture mechanics approach is used to relate the interface adhesion energy to the unstuck length. Here, we consider the gap separation at the left end of the beam and the point loading as two applied loads (one applied displacement and one applied force) that determine the energy release rate. The mechanical energy release rate for this system is equal to (e.g. Lawn, 1993): 1 dUM d 1 1 2 d kc G= = ± F2 + h ; (3) b ds 2b ds 2b ds where UM is the total mechanical energy,  is the compliance relating the applied point load to the resulting load–point displacement, b is the width of the beam, and kc is the stiHness relating the applied gap separation to the reaction force at the post. The ± sign accounts for the direction of the applied load; pressing the beam towards the substrate corresponds to a decrease in the energy release rate, while pulling away from the substrate increases the energy release rate. DiHerentiating Eq. (2) with respect to the load yields +

x03 (s3 + 3sx02 − x03 − 3x0 s2 ) : (4) 3EIs3 The proportionality constant between the reaction at the post and the gap separation is equal to the stiHness of a cantilever beam with zero slope at both ends, 7xed displacement at one end and a prescribed displacement at the other. Elementary beam theory yields 12EI kc = 3 : (5) s Upon combining Eqs. (2) – (5), this yields =

G=±

EIh2 1 x04 (s2 − 2sx0 + x02 )F 2 + 18 : 2 bEIs4 bs4

(6)

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With no external load, this equation reduces to the conventional result reported elsewhere (e.g. Mastrangelo and Hsu, 1992; de Boer and Michalske, 1999; Rogers et al., 2002). An estimate for the interface adhesion energy, i , is obtained by assuming G = i and measuring of the initial unstuck length, denoted as s = s0 . 2.1. Model for adhesive healing of the interface with G = i If one assumes that the unstuck region adjusts to maintain G = i , Eq. (6) provides the relationship between applied load and unstuck length. It is convenient to use the initial unstuck length (with F = 0) as a normalizing factor, such that changes due to point loading are more readily observed. The initial unstuck length in the absence of point loading is given by (from Eq. (6) with G = i and F = 0, or from de Boer and Michalske, 1999); 1=4  3Et 3 h2 s0 = : (7) 2i Using this as the basis for normalization, Eq. (6) becomes ± 1024P 2

X 4 (S 2 − 2XS + X 2 ) 1 + 4 = 1; S4 S

(8)

where S = s=s0 ; X = x0 =s0 , and P is the load normalized by the value required to zero the gap separation at the center of a 7xed–7xed beam of length s0 , given by P = s03 F=(16Ebt 3 h). For compressive loads (P ¡ 0), the application of Eq. (8) is equivalent to saying that the interface heals, in that the unstuck length decreases to maintain G = i . In traditional fracture problems, Eq. (8) is not applied for compressive loads (when G ¡ i ). However, healing scenarios may be possible for some adhesion mechanisms, notably those that involve gap-dependent attractive forces. In such cases, compressive loads that reduce in the gap separation may trigger re-adhesion at the edge of the adhered region. 2.2. Model for mechanical contact (without adhesion): G = 0 Naturally, rigorous solutions for compressive loading would involve considering both adhesion and mechanical contact. One simplistic approach modeling mechanical contact is to enforce G = 0, which corresponds to zero opening displacements at the original edge of the contact (i.e. the original unstuck length). In this case, the unstuck length for a given applied load is estimated by setting Eq. (6) equal to zero; after normalization, the result is 1 1 S= + X: (9) 32 PX 2 In this scenario, the unstuck length grows to in7nity as the applied load goes to zero, since there is no adhesion to keep the beam in the de8ected position. Note that this result does not truly depend on i , since s0 can be canceled from both sides.

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For an initially adhered beam, and assuming that the unstuck length changes are only due to mechanical contact, it is reasonable to assume that the unstuck length is 7xed (i.e. s = s0 or S = 1) until the compressive load closes the gap separation at the edge of the adhered region. After this point, load increases dictate decreases in unstuck length to enforce G =0. Put another way, the proposed model for contact assumes the unstuck length is the original length until the applied load closes the gap separation, de7ned as G = 0. After this point, Eq. (9) is applied to determine S (instead of Eq. (8)). These models are compared with experiments and 7nite element simulations without adhesion in Section 3.2 (look ahead to Fig. 5). 2.3. Application of the adhesion models with G = i In microscale measurements, the stiHness of the contact often proves a useful means for interpreting the load–displacement data obtained during experiments. We de7ne an eHective stiHness of the beam as the derivative of the relationship between applied load and load–point de8ection. If one allows the unstuck length to change during the loading (or unloading) process (say, to maintain Eq. (8) or (9)) this will involve two contributions. We de7ne the eHective stiHness as follows: 1 d 9 9 dS ≡ = + ; (10) K dP 9P 9S dP where  is the normalized load–point de8ection (given next as Eq. (11)). The 7rst contribution represents the instantaneous mechanical stiHness of the system, i.e. the slope of the displacement–load relationship (of the load point) without any changes in unstuck length. The second term represents the change in the load–displacement curve due to adhesion-induced changes in stuck length during loading. In terms of the normalized quantities introduced earlier, the de8ection of the beam at the load point is given by (x0 ) X 2 (64XPS 3 + 192SPX 3 − 64PX 4 − 2X − 192PS 2 X 2 + 3S) = : (11) h S3 Eq. (11) can be used to calculate the partial derivatives of , while the partial derivative of S is calculated from Eq. (8) or (9). Note that when  → 1, the beam bottoms out at X = 1. Also, for P = 0 Eq. (11) recovers the s-shaped pro7le originally given by de Boer and Michalske (1999). Fig. 2 shows a normalized plot of how the unstuck length S is aHected by the applied load for diHerent load locations, assuming the unstuck length changes to maintain G = i . A key feature is that for very small loads, the unstuck length does not change and the stiHness equals that of a 7xed–7xed beam of length s0 . For small loads the eHective stiHness is directly related to the initial unstuck length, from which the adhesion energy can be estimated. Thus, measuring the eHective stiHness of the beam provides an alternative to interferometric techniques for experimentally determining the adhesion energy. Although experimental results are not presented here for pulling on the beam, the model illustrates interesting behavior that motivates future experiments. When pulling on the beam near the support post, the displacement is dominated by the stiHness of ≡

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Unstuck length, S = s/so

1.5

1.25 Load location, X = xo /so = 0.1 0.3

1 0.9 0.7

0.75 PULL

0.5 -0.5

PUSH

0

0.5

0.5

1

1.5

2

Fso3 Applied point load, P = 192EIh Fig. 2. Predictions of the unstuck length vs. the applied point load for diHerent load locations, assuming the unstuck length changes to maintain G = i .

the short span between the load and the support. For this case, the crack length grows stably as the load increases. When the location of the pulling load is orientated closer to the edge of the crack tip, the crack grows unstably for a 7nite distance until it arrests. The transition between these two behaviors is determined by evaluating the stability of Eq. (6); the critical load–point location is found to be X ¿ (2=3)3=4 . Eqs. (8) and (11) enable estimates of the force or displacement required to free a stiction-failed cantilever beam, assuming the beam is in7nite in length. Fig. 3 illustrates the relationship between load and load–point displacement for the same cases shown in Fig. 2. The de8ection for P = 0 corresponds to the displacement from an initially freestanding beam to the s-shape. Varying degrees of sensitivity can be seen for the diHerent load locations. For each case the load–point displacement asymptotes to  = 1 as the (compressive) load increases, since the beam bottoms out at the load location. As an alternative to matching load–displacement curves, the eHective stiHness can be used to characterize the adhered con7guration. Fig. 4 illustrates the predicted stiHness as a function of an applied compressive load, assuming G = i is enforced. In the limit where the load goes to zero, the stiHness approaches that of a 7xed–7xed beam of length s0 . The initial dip in the stiHness seen in Fig. 4 can be explained as follows. In most fracture approaches, crack healing is usually prohibited (i.e. for G ¡ i the crack is not shortened to increase G and maintain the equality G=i ). However, decreasing G from the initial equilibrium point corresponds to smaller gap separations near x = s, and one may play the game that these smaller crack openings trigger re-adhesion. In this case, the increased adhesive contact required to maintain G = i causes the beam to de8ect further resulting in increased load–point de8ections. As such, the total compliance goes

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Load point deflection, ∆ = δ(xo)/h

-0.5 PUSH

Load location, X = xo/so = 0.1

0

0.3

PULL

0.5

0.5 0.7 0.9

1 -0.4

-0.2

0

0.2

0.4

Applied point load, P =

0.6

0.8

1

Fso3 192EIh

Fig. 3. Predictions of the load–point de8ection vs. applied load for diHerent load locations, assuming the unstuck length changes to maintain G = i ; the load–point bottoms out on the substrate when  → 1.

Load location, X = xo/so = 0.1

Stiffness, K =

kso3 192EI

25

20 0.9 15

10 0.7 0.5 5 0.3 0

0

0.2

0.4

0.6

0.8

1

3

Applied point load, P =

Fso 192EIh

Fig. 4. Predictions of the eHective stiHness vs. applied load for diHerent load locations, assuming the unstuck length changes to maintain G = i .

up, and the eHective stiHness decreases. At higher loads the stiHness increases because the rate at which the unstuck length shrinks is decreasing—i.e. ds=dP is small at high loads (see Eq. (10) and Fig. 2).

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3. Experiments 3.1. Overview The samples tested were polycrystalline silicon (polysilicon) cantilever beams fabricated at the Sandia National Laboratory using their 4-layer, SUMMiT process; complete details regarding fabrication are available in Rogers et al. (2002). The substrate surface that contacts the underside of the beams is comprised of the same polysilicon material as the beams. (Details regarding surface characteristics are given in Jones et al., 2003.) With regards to the release process, a 49% hydro8uoric acid solution etch was used to remove the sacri7cial tetraethylorthosilicate (TEOS) layers. The sample was rinsed in deionized water for 5 min and was then rinsed a second time in a fresh deionized bath for another 5 min. The 7nal rinse was in isopropyl alcohol for 5 min. Following the rinse cycle the sample was dried on a hot plate at 110◦ C for 10 min after removal from the isopropyl alcohol (Rogers et al., 2002). After the samples were dried, they were subjected to laser irradiation to repair the stiction-failed beams as part of a previous study (Rogers and Phinney, 2001). Following these tests, the beams have been exposed to ambient conditions for several months. The specimens’ dimensions were measured using a Zygo New View 5022 3-D surface pro7ler. Using scanning white-light interferometry, this system can measure sub-nanometer out-of-plane displacements. A 10 × objective was used on the microscope with an image zoom factor of 2. The deformed shape of the beams and their dimensions were determined from the average of 5 scans, where each scan was a 1 s; 5 m bipolar scan. The average dimensions of the beams were found to be: 30 m wide (b); 2:6 m thick (t), with lengths varying from 100 to 1500 m (L). The gap separation (h) between the beam and substrate was measured to be 1:85 m. The chip has arrays of beams with diHerent lengths; the history of the sample (i.e. release procedures, laser irradiation and storage) produced all three beam con7gurations shown in Fig. 1. An MTS nanoindentation system with a “dynamic contact module” (DCM) indenter head was used to actuate mechanically the micro-scale beams. The DCM is an ultra-low load indenter with load resolution in the tens of nano-Newton range and nanometer displacement resolution (under typical operating conditions). A sharp Berkovich indenter tip was used, resulting in point contact between the beam and probe. Plastic penetration of the probe into the beam can be quanti7ed by indenting an adhered portion of a beam to the same maximum load and unloading; for the cases considered, here, the plastic penetration of the probe under 2 mN (the maximum load considered) was ∼ 70 nm, which is approximately 5% of the total displacement of the beam. Penetration of the tip will be much smaller for smaller loads when the probe is placed on a freestanding portion of the beam; as such, displacements due to plastic penetration are neglected in the results presented in this section. All tests were conducted within an environmental isolation chamber at room temperature, which ranged from 19◦ C to 21◦ C. The relative humidity remained between 30% and 34% for all of the tests. For any individual test the relative humidity stayed within 1% of the initial value. Sequences of tests were conducted on multiple days

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over a period of several weeks; this introduced no readily apparent scatter in the data. When possible, standard deviations are shown with the data and are typically on the order of 5% of the measurement. The elastic modulus of the beams was measured using an experimental technique outlined by Weihs et al. (1988), which involves using a nanoindenter to displace the freestanding beam. The slope of the load–displacement curve is used to determine Young’s modulus. Multiple freestanding cantilever beams were loaded at several diHerent locations along the length of the beam; at each position, 10 tests were conducted and the load–displacement data were averaged. The tests were conducted under load-control at a rate of 1 N=s until a maximum de8ection of 500 nm was reached, and subsequently unloaded at the same rate. From this data, the 8exural rigidity for these beams was determined to be 7:264 N m2 with a standard deviation of 0:299 N m2 (±4:1%). Assuming that Poisson’s eHect is negligible for beams of these proportions (Jensen et al., 2001), the elastic modulus is calculated to be 164:7 GPa with a standard deviation of 6:8 GPa (±2:4%). This value is consistent with other published data on polysilicon beams of this scale (e.g. Jensen et al., 2001). 3.2. Adhesion energy estimates from s-shaped beams before and after point loading Prior to loading the beams with the nanoindenter, the de8ection pro7les for six identical s-shaped beams were measured using conventional interferometry. Each beam had a total length of 1000 m; the horizontal spacing between the beams was 8 m. The average unstuck length s0 was measured at 164:3 ± 9:7 m. Using the results developed by de Boer (and obtained from Section 2 with F = 0) this translates into an adhesion energy equal to 20:6±1:22 mJ=m2 (6%). This is approximately 20 times higher than previous results: both those determined from the same type of sample after laser treatment and prior to storage (Rogers and Phinney, 2001), and those determined from similarly fabricated beams (de Boer and Michalske, 1999). Possible reasons for this diHerence are discussed in the next section. Each of the six beams were subsequently loaded and unloaded 7ve times, each time near the center of their free length and to a maximum load of 2 mN. After loading, the unstuck lengths were measured via interferometry to be 152:1 ± 1:6 m. This length corresponds to a surface energy value of 27:9 ± 0:29 mJ=m2 (1.1%). It seems clear that actuating the beams reduces the variations in stuck lengths and signi7cantly increased the adhesion energy as inferred from fracture mechanics models. Fig. 5 compares the measured stiHness vs. load–point displacement relationship with the theoretical models for loads large enough to signi7cantly change the unstuck length. The beam is initially in the s-shape shown in Fig. 1c and remains as such for the duration of the test. The probe location is 75 m from the support post. Experimental stiHness values were obtained by numerical diHerentiation of each of six load–displacement measurements, which were then averaged. The error bars again correspond to the standard deviation from six tests; error bars smaller than the plotting points have been dropped for clarity. Both the loading and unloading portions of the response are included. At very small depths (. 100 nm), the contact area and corresponding contact stiHness is small and in8uences the results. At larger depths, the contact stiHness is an

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800

104

FEA 600

Γi = 34 Γi = 0

Stiffness, N/m

400

Γi = 28 mJ/m2 0

200

400

600

FEA (contact simulation) Γi = 0 (contact approx.)

103

Γi = 28x10-3 J/m2

Experiments, loading Experiments, unloading

0

200

400

600

800

1000

Load point displacement, nm Fig. 5. Comparison of predicted and measured stiHness as a function of load–point displacement.

order of magnitude larger than the structural stiHness of the beam and thus does not play a signi7cant role in the overall system stiHness. Using the stiHness from displacements in the ∼ 200 nm range (i.e. for small loads, K ∼ = 400 N=m), the initial unstuck length is estimated as s0 = 153 m. This is in agreement with interferometric measurements discussed above, and corresponds to i ≈ 28 × 10−3 J=m2 . The error bars again correspond to the standard deviation from six tests. Several theoretical models are imposed in Fig. 5. The solid line represents the case where G = i is maintained at all loads by shrinking the freestanding region. The long dashes represent the analytical contact model that 7xes the unstuck length until G = 0, and then shrinks the unstuck region to maintain this equality. Finally, the short dashes are the results of a 7nite element simulation with the following features: measured material properties, plane strain elements, a rigid substrate, and explicit contact modeling with zero friction (Begley and Murphy, 2003). For small applied loads (displacements), there is a slight drop in the experimental stiHness curve, as predicted by the adhesion model. This is a result of additional downward displacements that arise while decreasing the freestanding region (i.e. healing) to maintain G = i . This dip is highly repeatable: error bars (standard deviation) from six diHerent measurements are smaller than the plotting points and thus are not plotted. The relative size of the stiHness drop is not yet well established, as the initial stiHness is obfuscated by the establishment of contact between the probe and the top surface of the beam. The inset to Fig. 5 illustrates that a larger assumed value of adhesion energy (and thus shorter initial unstuck length) yields better agreement over the 7rst ∼ 400 nm of displacement. It also reduces the discrepancy between the 7nite element contact model and the model with G = 0. Using i = 34 × 10−3 J=m2 corresponds to an

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2500

Stiffness, k (N/m)

Unstuck length, so = 152 µm

2000

1500 0.95 so

1000

1.08 so

500 Measurements: marker dimensions are +/- standard deviation

0 0

20

40

60

80

100

120

140

Load location, xo (µm) Fig. 6. Comparison of the predicted and measured stiHness for several probe locations; the predictions are for unstuck lengths equal to initial unstuck length measured via interferometry, and values slightly above and below this value.

initial unstuck length of s0 = 144 m, which is about 5% shorter than that estimated via interferometry. To provide additional comparisons with the models outlined in Section 2, 7ve different positions along the length of each of the six beams were loaded mechanically at a loading rate of 10 N=s. The locations were 25; 50; 75; 100 and 125 from the supported end of each 1000 m beam. Again, the initial unstuck length was approximately 150 m. Fig. 6 compares the experimental stiHness vs. load location to the theoretical predictions for three initial unstuck lengths. The stiHness value chosen in Fig. 6 is that corresponding to small loads/displacements (i.e. F ∼ 0:25 mN; ∼200 nm). For small loads, the unstuck length does not change appreciably and the stiHness is simply that of a 7xed-7xed beam, determined via Eq. (11). The error bars represent the standard deviation from six tests. The results in Fig. 6 indicate that choosing i =34×10−3 J=m2 to improve agreement for small displacements (e.g., see the inset of Fig. 5) is in part inconsistent with other characterizations. This is discussed in greater detail in Begley and Murphy (2003). Most importantly, the eHective stiHness calculation for small loads/displacements is reasonably sensitive to the unstuck length. This indicates that favorable resolution is possible if one wishes to back out the adhesion energy without viewing the sample by slightly perturbing the system with small loads and calculating the system’s stiHness. The agreement is better at the locations further from the support post; for loading near the post, the compliance of the post in8uences the measured response (e.g. Jensen et al., 2001) and increases the discrepancy. 3.3. Transitions between adhered states during mechanical loading Transitions between the adhered states shown in Fig. 1 can be explored by loading an initially freestanding beam. Fig. 7 illustrates the result of this type of experiment by

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2.5 Load applied at xo = 50 µm Beam length L = 100 µm

Load, F (mN)

2

1.5

Loading 1

B Free-standing

Arc-shape

Unloading

0.5 C

A

S-shape

0 0

500

1000

1500

2000

Load point deflection, δ (nm) Fig. 7. Experimental load vs. load point de8ection for a short, initially freestanding beam; close-ups of the transitions at A, B and C are shown in Fig. 8.

plotting load–point displacement vs. load for a full load/unload cycle. The experiment is conducted in displacement control, wherein the applied load is continually adjusted to maintain a prescribed displacement rate (see Jones et al., 2003 for more details). The load–displacement relationship clearly illustrates three transitions. The beam’s response for small displacements is that of a freestanding cantilever beam; at point A ( ∼700 nm), the tip of the beam adheres to the substrate, and the response transitions to that of an arc-shaped beam. At point B ( ∼1300 nm), the beam transitions from the arc-shape to an s-shape, where it remains for the duration of the loading portion. As the probe is unloaded (i.e. allowed to release upwards, corresponding to decreasing beam de8ection), the beam transitions from the s-shape back to the arc-shape at point C ( ∼1250 nm). At this transition, there is a distinct jump in displacement at nearly constant load. Upon complete unloading, when the probe is pulled oH of the beam and the corresponding load goes to zero, the beam remains in an arc-shape. The 7nal displacement of the probe ( ∼550 nm) corresponds to the downward de8ection of the load point due to the transition from freestanding to arc-shaped. Details of these transitions are shown in Figs. 8a–c. In Fig. 8a, the increasing spacing of the data points at the transition from freestanding to arc-shaped re8ects a rapid change in beam position. During this transition, the feedback control loop is unable to respond quickly enough to maintain the prescribed displacement rate. (Further details of the measurements are provided in Jones et al., 2003.) This is evidence of attractive forces between the tip of the beam and the substrate, and may be evidence of a jump-to-contact or the spontaneous formation of a bridging capillary at the beam’s tip. Regardless of the mechanism, there is some type of attraction that slightly reduces the force required to maintain the prescribed displacement. Close inspection of the second

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0.65

0.128 Free-standing to arc-shape transition

0.124 0.120

(a)

Loading

Load, F (mN)

Load, F (mN)

0.132

650 660 670 680 690 Load point deflection, δ (nm) (b)

0.60 Arc-shape to s-shape transition

0.55 Loading 0.50 1200 1250 1300 1350 Load point deflection, δ (nm)

Load, F (mN)

0.41 Unloading 0.40 0.39 0.38

(c)

S-shape to arc-shape transition

1100 1140 1180 1220 Load point deflection, δ (nm)

Fig. 8. Enhanced view of the experimental load–displacement relationships during transitions in adhered geometry; the full load–displacement curve is shown in Fig. 7.

transition shown in Fig. 8b reveals a more continuous transition, during which the system stiHness increases. Upon unloading, the beam remains in the s-shape, with a load–displacement relationship that clearly diHers from the loading portion. The transition at point C reveals unstable behavior that is similar to predictions of adhesion in problems involving structures with 7nite stiHness (e.g. Attard, 2000). At ∼1190 nm, the probe experiences a rapid jump in displacement that corresponds to the beam lifting oH of the substrate (i.e. decreasing its downward displacement corresponds to the beam pushing the probe upwards). During this transition, the feedback system is once again unable to maintain the prescribed displacement rate (Jones et al., 2003). At the end of the transition, the load point has moved about 100 nm, and the load has increased slightly. This re8ects that the fact that part of the adhesion region pushes upwards on the probe, causing a slight jump in the force required to maintain the prescribed displacement. These transitions in adhered geometry are perhaps even more apparent in the relationship between the eHective stiHness of the system and load–point displacement, which is shown in Fig. 9. (Note that Fig. 9 corresponds to the same test shown in Figs. 7 and 8, such that x0 = 50 m and L = 100 m.) The jump from freestanding to arc-shaped is accompanied by a large jump in stiHness, as shown in Fig. 9 for ∼680 nm; the stiHness jump is in very good agreement with beam theory predictions for freestanding and cantilevers pinned at the tip. At approximately ∼1250 nm the beam’s stiHness transitions to that of an s-shaped beam. Upon unloading, the unstable transition back to arc-shape appears as a large dip in stiHness; in reality, the slope

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Maximum displacement (initial gap size): δ ~ 1800 nm

Stiffness, k (N/m)

4000 Unloading: s-shape

3000

2000

Unloaded: arc-shaped

1000

Loading: free- to arc-shape transition

Unloading: s- to arc-shaped transiton

Initial loading: free-standing

0 0

500

Loading: s-shape

1000

1500

2000

Load point deflection, δ (nm) Fig. 9. Experimental stiHness (calculated from measured load–de8ection data) vs. load–point de8ection for a short beam (i.e. x0 = 50 m and L = 100 m), illustrating the abrupt stiHness changes that occur during transitions in adhered geometry.

of the load–displacement curve is negative (see Fig. 8c), but numerical averaging of several load–displacement data points somewhat obfuscates this. Finally, one may note the very small stiHness observed for the early stages of contact. This eHect disappears once the contact size has increased to the point that the contact stiHness is much larger than the eHective stiHness of the beam—usually after the 7rst 30 nm of de8ection. 3.4. Cyclic loading: history dependence To explore the occurrence of load path dependence and the in8uence of loading rate in the experiments, tests were conducted wherein the applied load was ramped to a pre-set maximum and subsequently unloaded to zero. The tests described in this section were conducted on a 200 m initially freestanding cantilever beam loaded at 50 m from the support post (at room temperature with a relative humidity of 31%). Fig. 10 shows the load–displacement relationships from three of the 7rst 7ve load– unload cycles. The 7rst cycle is considerably diHerent from subsequent cycles, due to the shape transitions described in the previous section; the beam remains s-shaped for all future cycles. Repeated loading of the beam revealed that there is an initial shakedown period until the beam reaches a steady-state condition. After 7ve cycles the beam has reached a steady state wherein subsequent cycles follow the same loading–unloading hysteresis loop. The path dependence is clearly repeatable, even for cases that remain consistently in the same s-shape. Another way to illustrate this loading history is by considering the stiHness of the system for small loads, which is shown in Fig. 11. The stiHness increase for extremely small loads is a result of the growing contact size; after the 7rst 20 micro-Newtons or so, the contact stiHness is much larger than that of the beam, and the system responds

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2.5

Load location: xo = 50 µm Initial free beam with length: 200 µm Cycle 1

2

Load, F (mN)

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Cycle 2

Cycle 5

1.5 Loading

Displacement of load-point due to transition from free to adhered in in 1st cycle

1

Unloading

0.5

0

0

500

1000

1500

2000

Load point deflection, δ (nm) Fig. 10. Experimental load–displacement histories for successive loading of an initially freestanding beam; the beam transitions from freestanding to arc- to s-shape during the 7rst cycle, where it remains for subsequent cycles.

700 S-SHAPED ADHERED BEAM

600

Stiffness, k (N/m)

Cycle 5

500 400

Cycle 2

300 Cycle 1

200 100 0

FREESTANDING

0

0.05

S-SHAPED ADHERED BEAM

TIP STICKING: ARC-SHAPED

0.1

0.15

0.2

0.25

0.3

Load, F (mN) Fig. 11. Experimental eHective stiHness (calculated from measured load–de8ection data) of a 200 m cantilever loaded at 50 m, illustrating the transition from freestanding to arc- to s-shape during the 7rst cycle; the solid line is the beam theory prediction.

as though the contact were rigid. Superimposed on the data for the 7rst cycle are beam theory predictions using the dimensions of the beam and the experimentally determined modulus. For the s-shaped region, the stiHness is calculated using the assumption G = 0, although the results are nearly identical for all models when F ¿ 0:25 mN ( ¿ 600 nm) (see Fig. 5). Once again, note that during the 7rst cycle the beam

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Net energy change per cycle (pJ)

100 Cycle #1 transitions from free-standing to s-shaped

80

Permanent adhered area change: ∆E = Γi(b∆s) All other cycles s-shaped

60

40

20 Plastic dissipation

0 1

2

3 4 Cycle number

5

6

Fig. 12. Net energy change during load–unload cycle on a 200 m beam loaded at 50 m, as calculated from load–displacement measurements. The plastic dissipation contribution was calculated from repeated loading adjacent to the beam.

transitions from freestanding to arc-shaped to s-shaped. In this case, the transitions are smoother than for shorter beams, and the compliance of the support post has a larger eHect since the beam is being loaded relatively close to the post. Over the 7rst 7ve cycles, the stiHness of the system increases with each cycle, even for small loads. This represents a permanent incremental decrease in the unstuck length with each cycle. Again, part of the beam that is pushed into contact at higher loads remains adhered even after unloading. 3.5. Cyclic loading: energy dissipation The energy dissipated during each loading cycle can be estimated by calculating the area enclosed within the load–displacement curves; the results of this calculation are shown in Fig. 12 for the 7rst six cycles of actuation. Penetration of the tip of the diamond probe results in plastic deformation that contributes to the total energy dissipated. This eHect has been accounted for in Figs. 12–14 as follows. Multiple indents were performed on an adhered (fully contacting) section of the beam and the area enclosed by the load–displacement curve was calculated. The plastic dissipation due to repeated loading at the same location (in the adhered region) was measured for each successive load cycle, and decreases with each successive cycle as shown in Fig. 12. The steady-state energy dissipation due to plastic deformation under the indenter tip was ∼15 pJ ± ∼5 pJ for a maximum load of 2 mN. In Fig. 12, estimates of the energy dissipated due to plastic deformation and the growing of the contact area are superimposed. The estimate for the energy change associated with changing the unstuck length is taken to be i bSs, where Ss is esti-

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Net energy change per cycle (pJ)

100 5 µN/s

80 50 µN/s

60

10 µN/s

40

20

Maximum load: 2 mN Unstuck length: so = 152 µm

0 0

25

50

75

100

125

150

Location of point load, xo (µm) Fig. 13. Net energy change determined via load–displacement measurements (minus measured plastic dissipation) for diHerent load locations and three loading rates.

Net energy change per unit aarea, ∆E/(b∆s), mJ/m2

60

Maximum load: 2 mN Initial unstuck length: so = 152 µm

50 40 30

5 µN/s 50 µN/s

20

10 µN/s

10 0 0

25

50

75

100

125

150

Load location, xo (µm) Fig. 14. Net energy change normalized by the estimated change in the adhered area (via stiHness measurements) for diHerent load locations and measurement rates.

mated using the models presented in Section 2 with the condition G = i imposed. It is reasonable to conclude that the remaining energy dissipation is due to the bonding and debonding of the beam to the substrate as the load is cycled. The total energy change for the 7rst cycle shows a large contribution associated with the transition from the freestanding to s-shaped geometry.

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The change in unstuck length from cycle to cycle was inferred by considering the change in stiHness as illustrated in Fig. 11 and using the models described previously to calculate the unstuck length. Successive cycles show diminishing contributions from growing the adhered region until a steady-state condition is reached. The largest changes in energy dissipation occur during the 7rst few cycles, after which the behavior appears to be approaching a steady-state condition. To explore the nature of the dissipative mechanisms of the interface adhesion, stiction-failed beams were loaded at various load locations as well as diHerent loading rates. Fig. 13 shows the net dissipated energy (total energy dissipated minus the average energy dissipated from plastic deformation) for diHerent load locations and load rates. The error bars correspond to standard deviations from at least 7ve tests. The impact of loading rate is fairly inconclusive given the resolution of the energy calculation. The dominant factor of the energy dissipation appears to be due to the change in the adhered area rather than load rate. For a given load, the change in the unstuck length will be dependent on the load location, with the largest change when the beam is loaded near the center of the unstuck length. Similarly, the largest amount of energy dissipation occurs for loads at the center of the unstuck region. Fig. 14 plots the energy change normalized by the maximum change in surface area (SE=(bSs)) vs. load location, where Ss is the estimated change in adhered region that occurs in one cycle. This normalization illustrates that the energy dissipation scales roughly with the change in the size of the adhered region during a given load cycle. The uncertainty associated with the normalization increases as the load location moves towards the edge of the unstuck region, since the uncertainty associated with calculating Ss from measured stiHness increases in this regime. The agreement between this energy dissipated per unit area and the inferred adhesion energy supports the notion that the energy dissipation is directly related to the bond–debond process associated with the load cycle. 4. Discussion The use of an instrumented mechanical probe oHers several advantages for studying adhesion in microelectromechanical systems. In comparing experiments and theory, the system stiHness is favored over load–displacement relationships (such as those in Figs. 7 and 10), as it is less susceptible to eHects not accounted for in the modeling, such as penetration of the probe or rotation of the built-in end. As illustrated in Figs. 5, 6, 9 and 11, the adhered geometry can be directly related to measured stiHness of the beam. For s-shaped beams, this means that the interface adhesion energy can be inferred from the structural response of the component, as was done in Section 3.2. This approach broadens the scope of systems and environments that can be studied by eliminating the need to optically view the structure. A second clear advantage of using nano-mechanical probes is the opportunity to characterize transitions in adhered geometry in situ, as was done in Section 3.3. The exact details of the transition between freestanding, arc- and s-shaped beams provide unique experimental evidence (e.g. Figs. 7–9) that can be used to validate various

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adhesion mechanisms. The path dependence and unstable transitions shown in Figs. 7 and 8 are particularly promising for comparisons with models, since models developed for single asperity adhesion have predicted similar phenomena (Greenwood, 1997; Attard, 2000; Feng, 2000). To complement these comparisons, measurements with instrumented probes allow cyclic loading and direct characterization of the associated energetic changes, as presented in Section 3.4. Previous experimental approaches have not yielded the same rich information regarding history dependence (as shown in Figs. 10 and 11) and hysteresis or energy dissipation (as shown in Figs. 12–14). Here again, the additional information presented by mechanical probing provides opportunities for modeling to establish the mechanisms underlying adhesion. Appropriate models for adhesion should thus be capable of capturing a variety of behaviors, including transitions between adhered states, changes in adhered regions upon load cycling and hysteretic behavior. The wide range of processing and environmental conditions makes it implausible that a single mechanism is responsible for all adhesion observations in microscale components. This is illustrated by the fact the inferred adhesion energy in the present experiments on aged specimens is in the 20–30 mJ=m2 range, in contrast to experiments on similar samples immediately following the release procedure, where adhesion energies in the range 0:05–1 mJ=m2 . On-going mechanical loading experiments by the authors on recently released specimens reveal smaller measured stiHness, corresponding to interface adhesion energies in the 0:1 mJ=m2 range; this is in complete agreement with interferometric measurements on the same samples, as well as previous studies (e.g. de Boer and Michalske, 1999; Rogers and Phinney, 2001). DiHerent adhesion mechanisms are most likely responsible for the diHerence between just-released and aged samples. The initial (and apparently weaker) mechanism immediately after release may be van der Waals forces (e.g. de Boer and Michalske, 1999). Subsequent aging in ambient air may increase hydrocarbon surface contamination and make the surfaces more hydrophilic, increasingly the likelihood of capillary condensation and arguably leading to larger adhesion energies (Israelachvili, 1992). Alternatively, charge accumulation during handling (or subsequent probing as discussed in Section 3.2) may also increase electrostatic attraction. These mechanisms should be explored in terms of the ability of appropriate models to correlate with the types of measurements presented here (path dependence, adhesion transitions, energy dissipation, etc.). Elementary models which characterize adhesion purely in the context of an adhesion energy (such as those presented here) are capable of capturing only some of the behaviors illustrated here, notably the gross transition between adhered geometries. A fracture-based approach that does not account for contact and separation-controlled healing predicts stronger healing than reality (at least for small loads). Future modeling that includes mechanical contact and separation-dependent attractive forces is needed to capture the details of bond–debond transitions, load path dependence and cyclic behavior. Such models that do include an interaction distance (such as those developed for point-contact using Lennard–Jones-type potentials) predict behavior that is qualitatively similar to that shown in Figs. 6–12 (Greenwood, 1997; Attard, 2000; Feng, 2000). Similar models developed for the MEMS cantilever geometry will create

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attractive opportunities to directly evaluate proposed adhesion mechanisms using the experiments described here. Acknowledgements The authors gratefully acknowledge the support of the National Science Foundation, through award numbers NSF/CMS 0085122. We are particularly indebted to Prof. L. Phinney at the University of Illinois at Urbana-Champaign for helpful discussions, and very grateful for her group’s contribution of the MEMS cantilevers. References Abe, T., Messner, W.C., Reed, M.L., 1995. EHects of elevated temperature treatments in microstructure release procedures. J. Microelectromech. Systems 4, 66–75. Attard, P., 2000. Interaction and deformation of elastic bodies: origin of adhesion hysteresis. J. Phys. Chem. B 104, 10635–10641. Begley, M.R., Murphy, K.D., 2003. Determining adhered lengths and adhesion energy in MEMS cantilevers via structural stiHness measurements, to be published. de Boer, M.P., Michalske, T.A., 1999. Accurate method for determining adhesion of cantilever beams. J. Appl. Phys. 86, 817–827. de Boer, M.P., Knapp, J.A., Michalske, T.A., Srinivasan, U., Maboudian, R., 2000. Adhesion hysteresis of silane coated microcantilevers. Acta Mater. 48, 4531–4541. de Boer, M.P., Knapp, J.A., Clews, P.J., 2001. EHect of nanotexturing on interfacial adhesion in MEMS. International Conference on Fracture, Honolulu, Hawaii, December. Feng, J.Q., 2000. Contact behavior of spherical elastic particles: a computational study of particle adhesion and deformations. Colloids Surfaces A: Physiochem. Eng. Aspects 172, 175–198. Greenwood, J.A., 1997. Adhesion of elastic spheres. Proc. Roy. Soc. London A 453, 1277–1297. Israelachvili, J.N., 1992. Intermolecular and Surface Forces, 2nd Edition. Academic Press, Elsevier Science, Ltd., London, UK. Jensen, B.D., de Boer, M.P., Masters, N.D., Bitsie, F., LaVan, D.A., 2001. Interferometry of actuated microcantilevers to determine material properties and test structure nonidealities in MEMS. J. Microelectromech. Systems 10, 336–346. Jones, E.E., Murphy, K.D., Begley. M.R., 2003. Mechanical measurements of adhesion in micro-cantilevers: transitions in adhered geometry and cyclic energy changes. Exp. Mech., to appear. Lawn, B., 1993. Fracture of Brittle Solids, 2nd Edition. Cambridge University Press, New York. Mastrangelo, C.H., Hsu, C.H., 1992. A simple experimental technique for the measurement of the work of adhesion of microstructures. IEEE Conference Proceedings, Hilton Head. Rogers, J.W., Phinney, L.M., 2001. Process yields for laser repair of aged, stiction-failed, MEMS devices. J. Microelectromech. Systems 10, 280–285. Rogers, J.W., Mackin, T.J., Phinney, L.M., 2002. A thermomechanical model of adhesion reduction for MEMS cantilevers. J. Microelectromech. Systems, submitted. Weihs, T.P., Hong, S., Bravman, J.C., Nix, W.D., 1988. Mechanical de8ection of cantilever microbeams: a new technique for testing the mechanical properties of thin 7lms. J. Mater. Res. 3, 931–942.