Effect of Sacrificial Layer Compliance on Thermal Stresses of Composite MEMS Beams Jin Qiu, Senior Member, IEEE, Shawn Cunningham, Arthur Morris, Senior Member, IEEE Abstract — We present a detailed analysis of thermal mismatch stress (TMS) [1] developed in composite beams built on compliant sacrificial layer (SAC) [2]. Strikingly, the stress-balance of beams is strongly affected by the SAC compliance, which is substantiated by comparison of analytic models, finite element analysis (FEA), and measurements. With this analysis we concluded that the highly compliant organic SAC was a root cause for high and unequal biaxial curvatures in early devices. By switching to a rigid SAC we verified that the beam curvatures were significantly reduced.

(6) Ox is patterned and SAC is removed to free the beam. Several other process steps and beam layers are necessary for process integration and device functionality but are not important for stress balance. Thus we omit them in the process description.

Keywords — composite beam curvature, metal plasticity, sacrificial MEMS process, switched capacitor, thermal residual stress

I. INTRODUCTION We are motivated by maximizing the on-state capacitance of a switched capacitor with a flat fixed plate on substrate and a movable plate fabricated by a sacrificial MEMS process. This switched capacitor is a parallel plate capacitor with a tunable gap. In the off-state, the capacitor has an as-fabricated gap defined by sacrificial layer (SAC); and in the on-state, the gap is closed by electrostatic actuator.

Fig. 1: Key layers and process steps for moving capacitor plate. The movable plate is a metal-oxide-metal composite processed on a low-modulus, organic SAC. The key beam process sequence [3] is shown in Fig. 1. (1) The starting substrate contains CMOS layers and the lower capacitor electrode (not shown); (2) an organic SAC layer is deposited on substrate and patterned. (3) a metal layer M1 is deposited at Tm≈325ºC on SAC and then patterned; (4) an oxide layer (Ox) is deposited at Tx ≈ Tm+50ºC but not patterned; (5) a metal layer M2 is deposited on Ox at Tm and then patterned similarly to M1; and

This paper is based on RF MEMS development work in WiSpry, Inc. Jin Qiu is with Spatial Photonics, Inc, Sunnyvale, CA, USA. He can be contacted by E-mail: [email protected]. Shawn Cunningham and Arthur Morris are with WiSpry, Inc, Irvine, CA, USA.

Fig. 2: Wyko 3D image of cantilevered beam shape monitors. M1/M2 were designed with the same thickness and area. They also have equal deposition temperatures, which are conventionally considered as stress-free-temperature (SFT). Normally we expect a flat moving plate because of the high symmetry. However, early devices showed large curvatures of the movable plate. Beam shape monitors were measured by a white light interferometer as shown in Fig. 2. The kx/ky (short/long side) exhibit asymmetry with mean values of roughly -300/m and +100/m, and standard deviation of 100/m and 50/m, respectively. ky also showed a length and position dependence. Such beam shape resulted in low on-state capacitance, which prompted us towards an in-depth process analysis of thermal stresses. We have determined that SAC flatness contribution to beam curvatures is small. Intrinsic stress related to the growth and crystallization of thin films may also play a secondary role to the curvatures, but is not the subject of this paper. In the following sections, we describe the full sequence of beam shape analysis. First, we present an approximate formula linking capacitance and curvatures. Second, we summarize the composite beam curvature model on layer properties. Third, we analyze the constrained deformation of a single layer film on compliant SAC, with which we illustrate the effect of SAC compliance on film stress. Fourth, we describe the elastic beam process model implemented in FEA, which shows how stress imbalance results from a seemingly symmetric design and process. Fifth, we describe the metal plasticity effect on the layer stresses, and how it is influenced by SAC compliance. Sixth, we show the experimental verification of the process model developed. And last, we summarize the key conclusions.

II.

CAPACITANCE MODEL

The flat fixed plate and the movable plate have the same rectangular size of 2Lx by 2Ly. The theoretical maximum on-state capacitance is expressed by Eq. 1, where ε0 is the air permittivity, gd is the dielectric layer thickness, εd is the dielectric constant, and g0 is the equivalent air gap of the dielectric layer.

C max =

Con Cmax

ε dε0 A gd

=

ε0A g0

, where g 0 =

gd

εd

(1)

( )

+ ⎧ 1 l g0 g 0 arctan K g dx = ⎪ 2 + ∫ 0 kx l g 0 + g1 Kg ⎪ g 0 + g1 + ⎪ 2 =⎨ arctan K g− g0 g0 ⎪1 l dx = 2 2 ⎪ L ∫0 + − g g kL 2 K g− k (L − x ) 0 1 g 0 + g1 + ⎪ ⎩ 2

( )

where K g+ =

kL2 2 , K g− = g 0 + g1

k>0

(2) k<0

kL Con g0 = , where g k ≡ ck Cmax g0 + g1 + g k 2 g k = ckx

k x L2x 2

+ cky

k y L2y 2

⎧0.3 k x , y > 0 , ckx, ky = ⎨ ⎩0.6 k x , y < 0

(a) Positive curvature

With this model, we specified the maximum curvature tolerance to meet our capacitance requirement. We can make capacitance or curvature induced gap contours (Fig. 16) in a kx/ky plot. The target curvature and variability is much lower than what is shown in Fig. 2. We also conclude that if with the same absolute value, a positive k with a lower ck is preferred over a negative k. III. MULTILAYER CURVATURE MODEL

− kL2 2 g 0 + g1 − kL2 2 2

effect into a coefficient ck and the maximum curvature induced gap kL2/2. Fig. 4 plots ck as a function of (g0+g1) and k at 65 µm electrode length, which shows that ck can be approximated by two distinct values depending on the sign of k. Note ck also depends on L, but can be considered constant within a small range of L. Eq. 4 extends the results of Eq. 3 and Fig. 4 to an equivalent biaxial curvatures (kx and ky) induced gap. Numerical double integration confirmed that Eq. 4 is a good approximation.

(3) (4)

We use the same derivation approach in [1] but extend the solution to beams with variable modulus through out the thickness. As shown in Fig. 5, the layer strain is separated into two parts: the residual strain from process; and the deformation strain that is the sum of a reference strain at z = 0 and a curvature dependent strain proportional to z. Eq. 5 expresses the stress. Eq. 6 is the force balance, and Eq. 7 is the moment balance. By substituting Eq. 5 into Eq. 6 and 7, we solve for k in Eq. 8, which can be implemented easily in a spreadsheet or other software.

(b) Negative curvature

Fig. 3: 1D capacitance model on curvature and contact gap.

Fig. 5: Curvature/stress model for arbitrary number of layers.

σ ( z ) = E ( z )(ε m ( z ) − kz + ε r )

(5)

h

∫ σ ( z)dz = 0 ∫ σ ( z ) zdz = 0

(6)

0 h

(7)

0

h h Edz∫ Ez2dz − ⎛⎜ ∫ Ezdz⎞⎟ 0 ⎝0 ⎠

2

Fig. 4: Approximation of ck as a function of k.

h ⎡ ⎛ Ezdz ⎞⎟ ⎤ h ∫ m⎜ ⎢ ⎥ 0 k = ∫ Eε ⎜ z − h ⎟dz⎥ ⎢0 ⎜ ∫0 Edz ⎟⎠ ⎦⎥ ⎝ ⎣⎢

Two major factors that derate Cmax by preventing full contact of the electrodes are shown in the capacitance model in Fig. 3. First, the top electrode can have positive curvature (k>0, contact at center) or negative curvature (k<0, contact at edges). Second, the two electrodes can be separated at contact by a gap g1, due to electrode roughness, electrode hillocks, and/or foreign particles. Eq. 2 expresses the derating factor by g1 and k. Eq. 3 lumps the k

The curvature model suggests large stress imbalances to cause measured curvatures shown in Fig. 2, thus indicating an unknown cause for stress imbalance. Base on this model, we may modify layer stresses and thickness to tune the curvatures. Such tuning does result in some improvement, but does not help equalize the highly differentiated biaxial curvatures. Thus, we need to explore more on the root cause of beam curvatures.

∫

h

0

∫

h

0

(8)

Edz

IV.

SAC COMPLIANCE EFFECT ON SINGLE LAYER FILM STRESS

Before going to the composite beam analysis, let us first look at how the thermal deformation of a single layer film can be affected by SAC compliance. As shown in Fig. 6, we consider a one-degree-of-freedom single metal layer M1 (thickness, t1; modulus, E1; thermal strain, εT, length 2L) constrained by SAC (thickness, t2; shear modulus, G2 , low thermal stress neglected). The origin is centered on M1 with x denoting distance from the origin, and v denotes M1 lateral displacement away from origin.

listed in Fig. 7 caption, we have τ = 20µm.

ε = σ E1 = (dv dx − ε T ) E1t1 dε dx − (G 2 t 2 )v = 0

(10)

v x = 0 = 0, ε

(11)

=0

v (ε T τ ) = sinh(x τ ) cosh(L τ ) ≈ exp(( x − L ) τ ),

(12)

where τ = E1t1t 2 G2 σ (− E1ε T ) = 1 − q( x), where q( x) = cosh (x τ ) cosh (L τ )

σ

Fig. 6. Lumped stiffness model of a single layer M1 on SAC. (The vertical spring represents lateral shearing stiffness)

x=L

(9)

x =0

(− E1ε T ) = 1 − Q , where Q = 1 cosh (L τ )

(13) (14)

The derivation above implies two behaviors observed in Fig. 2. 1. Position dependence of the stress: As shown in Eq. 13 and Fig. 8(a), SAC provides the least constraint for metal near its edge, but this constraint increases exponentially away from metal edge. 2. Length dependence of the stress: As shown in Eq. 14 and Fig. 8(b), the maximum constraint of the SAC at M1 center is also a function of metal length. When the metal piece has different length and width, the constrained stresses developed will be different for the two orientations.

(a) Position dependence of stress relaxation factor

Fig. 7: Metal displacement, analytical model vs FEA. Modeled quarter beam has 90 µm length and 30 µm width. εT=0.6%. t1 = 0.5µm, t2 = 2µm, E1 = 70GPa, G2 = E2/2(1+0.3) = 0.2GPa M1 strain is expressed by Eq. 9. The force balance and boundary conditions are expressed by Eq. 10 and 11. Eqs. 9-11 are solved for the displacement v (Eq. 12), which matches well with FEA results as shown in Fig. 7. Both dv/dx and M1 stress σ (Eq. 13) are functions of x. Presented in Eq. 14, M1 stress at x=0 differs from its thermal stress E1εT by a ratio of Q, which is the unit-less figure-of-merit of SAC compliance (Q=0 is rigid, Q=1 is fully compliant). As plotted in Fig 8(b), Q decreases with the increase of the ratio of L over the characteristic length τ. With parameters

(b) Length dependence of stress relaxation factor Fig. 8: Position and length dependence of SAC stress relaxation factor. (factor of 0 is fully rigid, factor of 1 is fully compliant)

V. ELASTIC ANALYSIS OF FULL BEAM PROCESS With the insight gained from last section, we come back to explain the effect of compliant SAC on a full composite beam. Fig. 9 illustrates the thermal stress history of both M1 and M2 layers. M1:1 is the starting point for M1, where its stress is zero at deposition temperature Tm. When the temperature is cooled down to room temperature (RMT) and heated beyond Tm to the Ox deposition temperature Tx, M1 stays on the dashed curves with slope A. Referring to Eq. 13, slope A is proportional to the reduced stiffness of M1 due to compliant SAC. After the Ox deposition step M1:3, M1 is well constrained by the Ox. Ox has similar modulus as M1, but several times thicker, and is fully anchored to substrate. Ox also has a negligible CTE. So M1 will stay on curves with slope B (proportional to M1 stiffness) for all subsequent steps that may include several cycles of annealing. M1 will end up at the point M1:4. In contrast, M2 is deposited on the rigid Ox, so it will always stay on curves with slope B. Thus the final point of M2 is at M2:2. The M1/M2 stress imbalance is (slope B – slope A)*(Tx-Tm). The length and position dependence of curvatures is expected with the insight gained by Eq. 13-14 and Fig. 8.

There are two primary ways to suppress the final stress imbalance. 1. By making the SAC rigid, slope A will equal slope B. 2. By making Tx = Tm, M1:3 will be aligned with M2:1. However, introducing plastic analysis of the metal stress in next section will reveal that making Tx = Tm is not a valid option. The beam deposition process is modeled using finite element analysis (FEA) by sequentially adding/removing layers and importing stress conditions from previous layer stack analysis. This type of analysis can be implemented by “Import analysis” within Abaqus [4]. Fig. 10 illustrates the model of a quarter beam structure at the step before Ox patterning. VI. METAL PLASTIC ANALYSIS Deposited at high temperature of 325ºC and cooled down to room temperature, metal stress would reach more than 600MPa under elastic assumption. We have measured the metal stress using the wafer bow measurement of a monitor wafer as shown in Fig. 11. This showed the as-deposited stress was approximately 150MPa and relaxed to approximately 120 MPa. This evidence supports the yielding of the metal that is deposited at 325ºC. 160

Stress (MPa)

150 140 130 120 110 100 0

20

40

60

80

100

120

140

160

180

Time since deposition (hrs)

Fig. 11: Wafer bow measurement post-deposition and as a function of time. Fig. 9: Schematics of the thermal stress evolution of M1 and M2.

Fig. 10: FEA model of quarter composite beam structure on SAC. Structure bottom is fixed, and symmetric boundary conditions are applied to faces.

of

metal

stress

Several concepts of metal plasticity behaviors are summarized below. 1. Hydrostatic stress: When the material has equal normal stresses on three axes, it is under hydrostatic stress, which does not induce yield behavior. 2. von Mises stress: von Mises stress is formulized to represent an equivalent tensile stress that drives metal yield. When the three normal stresses are not equal, a non-zero von Misses stress results, and if it is above a threshold (yield stress), the metal will be in plastic region. The metals in our beam process are not constrained (zero stress) in the z direction, so we expect high von Mises stress. 3. Perfect plasticity: In perfect plasticity, the stress reaches a constant value at and above the yield strain.

4. Isotropic plasticity: In isotropic plasticity, stress continues to increase above yield strain, but with a reduced slope compared to the elastic region. In other words, the yield stress increases with plastic strain (strain hardening). This strain hardening effect is symmetric, i.e., when the metal strain reverses to the opposite sign, the increased yield stress remains. 5. Kinematic plasticity: Kinematic plasticity is similar to “isotropic plasticity”. But in this case when the strain is reversed, the yield stress is reduced typically by similar amount that it is increased beyond yield point initially. This is also called the “Baushinger effect” [5]. 6. Temperature dependence: Yield stress and strain can depend highly on temperature. Typically, the higher the temperature, the lower the yield stress. 7. Hybrid plasticity: Real plasticity is complex and can comprise multiple behaviors. Some possible stress-temperature loops are shown in Fig. 12.

include the isotropic plasticity assumption. Organic SAC has a glass transition temperature, Tg, on the order of 250 ºC, which is indicative of a transition in material properties. At temperatures above Tg, the modulus of the organic SAC material is a fraction of its value below Tg. FEA can model such temperature dependent SAC modulus as shown in Fig. 14(b), which is a very different beam shape than with a constant low modulus assumption as in Fig. 14(a). This demonstrates the importance of modeling and measuring temperature dependent parameters.

400 300

stress MPa

200

Fig. 13: Dependence of biaxial curvatures on SAC modulus. Metal has 280MPa yield stress, and isotropic plasticity [4] of 600MPa stress at 0.4% yield strain. Dimensions of the beam layers are the same as in Fig 7.

100 0 -1 0 0 -2 0 0

0

100

200 300 t e m p e ra t u re C

400

500

Fig. 12: Example metal stress-strain loops used to study the effects of plasticity on stress balance. Several plasticity assumptions were applied to rigid SAC simulations, all of which resulted in very low final curvatures. We believe the reason is that after the first couple thermal cycles, metal stress-strain curve will lock into a repeating loop, regardless of which plastic behavior it has. In the case of rigid SAC, M1 goes through two more cycles upfront than M2, but both will then go through several more cycles together which results in very similar final stresses. If the SAC is compliant, M1/M2 stress imbalance will result due to plasticity, even if we set Tx = Tm in simulation. Roughly the imbalance due to plasticity is similar to the imbalance due to (Tx-Tm) under elastic model. We believe the reason is that M1 goes through its first two thermal cycles with lower constraint than M2, this difference will result in a different final repeating stress-strain loops of the two metals. Fig. 13 shows the dependence of biaxial curvatures on SAC modulus. Minimized and equalized biaxial curvatures are achieved with SAC modulus higher than 5GPa. The results

(a) Constant SAC modulus of 0.5GPa.

(b) Temperature dependent SAC modulus: 5GPa at 23 ºC, 0.5GPa at 325 ºC. Fig. 14: FEA results of quarter beam. Short side has negative curvature, while long side has positive one.

VII. EXPERIMENTAL VERIFICATION

VIII. CONCLUSIONS

As verification, a rigid-SAC process was implemented and successfully demonstrated beams with significantly reduced curvatures. Typical beam shape is shown in Fig. 15. The cantilever beams B5 and B8 are comprised of a symmetric M1 and M2 that show a small deflection of 0.13-0.14 µm over a length of 150 µm. Beam B5 and B8 can be compared to Fig. 2, but the other beams have a different construction and should not be compared to Fig. 2. The range of curvatures for beams produced on the rigid-SAC layers are compared with those produced on compliant-SAC in a curvature induced gap contour plot, as shown in Fig. 16. The contours are made from Eq. 4, with 2Lx = 65 µm and 2Ly =150 µm. The curvature induced gap has decreased by about an order of magnitude with rigid SAC. This is significant for performance and yield of the devices. 0.14um 1.45um

0.35um

0.13um

B8 50 µm

B7 B6

This paper summarized our beam shape development work to reduce curvatures and to maximize on-state capacitance. In order to understand the whole sequence of beam shape effects, we developed the following analysis. 1. A process stress model, with the layer deposition sequences, thermal history, and material properties including plasticity as the model inputs. 2. A formula predicting curvature of composite beam with arbitrary layers of films, each layer can have different modulus and residual stress. 3. A capacitance model approximating the curvature effect by a curvature induced gap. The main conclusions of this paper are: 1. SAC compliance is an important factor in the resulting as-fabricated beam shape. 2. SAC material properties should be selected for a specific fabrication process and material system. 3. Metal plasticity behavior and process temperatures are important factors to consider for optimal beam shape. 4. Metal plasticity is less important when fabrication is on a rigid SAC layer. 5. Modeling (FEA or analytical) are important for estimating the effect of material properties and process sequence on the shape of a MEMS device.

B5 REFERENCE [1]

Fig. 15: Beam shape monitor with new rigid SAC process, measured by a Wyko white light interferometer.

[2] [3] [4] [5] [6]

Fig 16: Comparison of typical beam curvatures of rigid SAC process vs. compliant SAC process, in a curvature induced gap contour plot. Error bars indicate +/- 1 standard deviation.

T. G. Bifano, et. al., "Elimination of Stress-Induced Curvature in Thin-Film Structures," Journal of Microelectromechanical Systems, [11], no. 5, pp 592-597, October 2002. J. R. Stanec, et. al., “Realization of low-stress Au cantilever beams,” Journal of Micromechanics and Microengineering, [17], pp. N7-N10, 2007. S. Cunningham, et. al., “High-performance integrated RF-MEMS: Part I-the process,” European Microwave Conference, 2003. Abaqus Analysis User’s Manual, version 6.2 – 6.8. Y. Xiang, et. al., “Bauschinger and size effects in thin-film plasticity,” Acta Materialia, [54], pp. 5449-5460, 2006. Y-L. Shen, et. al., “Elastoplastic deformation of multilayered materials during thermal cycling,” Journal of Materials Research, [10], pp. 1200-1215, 1995.