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Temperature-dependent yield effects on composite beams used in CMOS MEMS
View the table of contents for this issue, or go to the journal homepage for more 2013 J. Micromech. Microeng. 23 035023
(http://iopscience.iop.org/0960-1317/23/3/035023)
J. Micromech. Microeng. 23 (2013) 035023 (8pp) doi:10.1088/0960-1317/23/3/035023
Temperature-dependent yield effects on
composite beams used in CMOS MEMS
F Y Kuo
1,2, C S Chang
1,3, Y S Liu
1,2, K A Wen
1,2and L S Fan
1,2,41Rm. 316, Microelectronics and Information Systems Research Center, No.1001, Daxue Rd., East Dist.,
Hsinchu City 300, Taiwan, Republic of China
2Institute of Electronics, National Chiao Tung University, Hsinchu, Taiwan, Republic of China 3Global Sensing Core, Inc., Hsinchu, Taiwan, Republic of China
4Institute of Nano Engineering and MicroSystems, National Tsing Hua University, Hsinchu, Taiwan,
Republic of China
E-mail:[email protected],[email protected](C S Chang),[email protected]
(Y S Liu),[email protected]@ieee.org Received 6 August 2012, in final form 16 January 2013 Published 1 February 2013
Online atstacks.iop.org/JMM/23/035023
Abstract
This paper presents an experimentally verified analytical model of temperature-dependent yield effects on the curvatures of composite beam structures used in complementary metal–oxide semiconductor microelectromechanical systems (CMOS MEMS). The temperature-dependent effects on composite beam curvatures of a thermal process can be predicted by extracting key parameters from the measured curvatures of a limited number of CMOS MEMS composite-layer combinations. The effects due to thermal history in MEMS packaging, which change the characteristics of beam curvatures due to material yield, are further analyzed. The models are verified with measured results from beam structures fabricated by an application-specific integrated circuit-compatible 0.18μm 1P6M CMOS MEMS process using a white light interferometer. These models can be applied in electronic design automation tools to provide good prediction of temperature-dependent properties related to CMOS MEMS beam curvature, such as sensing capacitance, for monolithic sensor system on chip design.
(Some figures may appear in colour only in the online journal)
1. Introduction
Sensor integration has attracted significant attention in recent years for enabling the sensing and processing of multiple environmental intelligences on a single electronics device. One integration approach is to integrate the complementary metal–oxide semiconductor (CMOS) circuit and microelectromechanical systems (MEMS) sensors on a compact monolithic substrate [1, 2]. Monolithic integration of these MEMS structures with circuits by the CMOS MEMS process may reduce overall chip size and avoid non-reproducible parasitic components and additional signal losses due to interconnection between the sensor and the circuit. Post-CMOS MEMS processing is commonly adopted and made compatible with the conventional CMOS application-specific integrated circuit (ASIC) process [3–5]. For these CMOS MEMS processes, however, MEMS structures made up of the
composite layers of metal and oxide experience temperature-dependent deformation due to residual stresses and variation of thermal stresses. The cantilever beam element, for example, is widely used in MEMS sensors and actuators including sensing fingers of the MEMS accelerometer [6], radio frequency (RF) MEMS switch [7], etc. The variation of the curvature of beams in these MEMS sensor designs may cause variation of sensing capacitance, which brings uncertainty to sensor read-out circuit design specifications and overall microsystem performances. The complicate residual stress distribution within composite layers at a given operation temperature makes it difficult to predict beam curvatures accurately in CMOS MEMS processes. To describe the composite-layer behavior, several analytical formulas have been discussed [16] for accurate modeling of elastic deformation of MEMS structures due to residual stress, from simple structures [14,15] to multilayer structures, including the extension of
J. Micromech. Microeng. 23 (2013) 035023 F Y Kuo et al the Stoney formula [18] and analysis based on the continuity
of strain among layers [19]. These works can be used to develop test patterns to extract residual stresses of fabricated multilayer MEMS structures [17]. However, temperature dependence for deformation modeling was not considered in these works. On the other hand, generalized formulas [8] and matrix forms [9] for curvature radius and layer stresses modeling caused by thermal strain in semiconductor multilayer structures have been developed. The analysis of temperature dependence of a three-metal CMOS process by introducing the coefficient of thermal expansion (CTE) is studied [10]. Extended modeling and validation for large-displacement beam actuator applications based on the matrix forms have also been proposed [11]. In this paper, we analyze and provide the temperature-dependent analytical model of the beam curvature of all the different layer combinations allowed by a complete ASIC-compatible CMOS process. The numerical values of beam curvatures can be predicted by key parameter extraction from the experimental data. The model covers the packaging thermal effects of the MEMS capping process on the beam curvatures due to material yield. This model can be used in computer-aided design tools to provide good prediction of temperature-dependent properties related to CMOS MEMS beam curvature, such as sensing capacitance, needed for the sensor system on chip (SOC) design. We also validate the analytical model with measurement results for all the different oxide–metal-layer combinations using a 1P6M 0.18μm ASIC-compatible CMOS MEMS process.
2. ASIC-compatible 1P6M CMOS MEMS process
A complete ASIC-compatible 1P6M CMOS MEMS process includes the foundry standard 1P6M CMOS process and the MEMS post-CMOS micromachining process. The microstructures are constructed by a dry-etch-based post-process. First, a hard mask (HM) layer is deposited on the standard CMOS wafer to define the MEMS structure. Second, a thick photoresist is coated to protect the non-MEMS area. Third, the microstructures are defined by anisotropic reactive ion etch of dielectrics. Fourth, an isotropic silicon undercut process is adopted to release the microstructures. Figure 1shows the sectional view of CMOS MEMS process flow step by step. The movable structures are made from the stack of interconnect layers in conventional CMOS technology. Different metal layers can be electrically connected by using via in IMD. Additional metal layer is utilized as an HM layer to define high aspect ratio microstructures from damage. The post-process achieves high aspect ratio structures with excellent flexibility of wiring.
The standard CMOS process allows combinations of the presence of six-metal layers, M1–M6. To observe the characteristics of the beam curvature of the process, cantilever finger structures with 25–32 different combinations of M1–M5 metal layers are observed at 9 different zones on test keys. Top metal layer M6 is preserved to improve the reliability of measurement using a Zygo white light interferometer.
The packaging process is necessary for the MEMS device to protect microstructures. Wafer level capping by glass frit is
(a) (d)
(b) (e)
(c) (f )
Figure 1. Sectional view of the CMOS-MEMS structure.
adopted in this process. The silicon cap wafer is pre-etched to reserve the space of the MEMS device, and then glass frit is printed on the bond ring by screen printing. After the soft cure process, the cap wafer is placed on the CMOS wafer with precise alignment, and then wafer to wafer bonding gets completed during stress and the hot cure process. The temperature of hot cure is about 350◦C. The capping process is shown in figures1(e) and ( f ). It is worth noting that glass capping is adopted in the experiment to allow the white light interferometer to measure the beam curvature through the transparent glass caps.
3. Analytical model for beam curvature
Consider a multilayer cantilever structure with N layers. Each layer i has a set of process/material properties Xi = {ti, αi,
σi, Ei} where tiis the layer thickness,αiis the temperature expansion coefficient,σiis the residual stress under reference temperature Toand Eiis the effective Young’s modulus. For a given metal–oxide combinations, the layer properties of all N layers can be denoted as X= {X1, . . . , XN}.
3.1. Calculation of radius of curvature
We derived the out-of-plane curl due to stress gradient in the cantilever as follows. The stress gradient produces the deformation with the radius of curvatureρ. The force along the z-axis (perpendicular to the curling beam layer) can be expressed as
f (z) = (σ (z) + εE (z)) w dz, (1)
whereε is the internal strain of the curling beam generated to cancel the force of gradient residual stress.σ(z) and E(z) are defined piecewise as
E(z) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ E1, z t1 E2, t1 < z t1+ t2 .. . Ei, i−1 tk< z i tk σ (z) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ σ1+ α1E(z)T, z t1 σ2+ α2E(z)T, t1< z t1+ t2 .. . σi+ αnE(z)T, i−1 tk< z i tk , (2)
whereT = T − Tounder the temperature T.
The net force for the beam shall be zero. Therefore,ε can be derived from the following equation:
t o (σ (z) + εE(z)) dz = 0 ⇒ ε = − t o σ (z) dz t o E(z) dz = −S/E, where S= t o σ (z) dz, E = t o E(z) dz. (3)
The neutral axis z= zoof a composite-layer beam shall meet
the following condition: t o E(z)(z − zo) dz = 0 ⇒ zo= t o E(z)z dz t o E(z) dz = Ez/E, where Ez= t o E(z)z dz. (4)
Considering the curling beam due to the residual stress, the radius of curvatureρ shall meet the zero-moment criteria:
t o σb(z − zo) dz = 0, where σb= (σ (z) + εE(z)) −E(z) ρ (z − zo) . (5)
Here,σbis the internal stress distribution as a function of z. Based on (3) and (4), we may rewrite (5) and derive the beam curvature 1/ρ: Sz− Ez ES− Ez2 ρ = 0 ⇒ ρ1 = 1 Ez2 Sz− Ez ES , where Sz= t o σ (z)z dz, Ez2 = t o E(z)(z − zo)2dz. (6)
Table 1. Oxide and metal layer parameters for the simplified model.
Xo Xm
to 0.8μm tm 0.58μm
σo(extracted) 100 MPa σm(extracted) −110 Mpa
Ey1 65 GPa Ey2 69 Gpa
αo 8.5e−6/◦C αm 2.3e−5/◦C
3.2. Simplified model and parameter extraction
For convenience, we define a binary coding rule to represent a given layer combination i as RM1M2M3M4M5M6, where Mx is 1 if layer x metal exists, and zero otherwise. For example, R000001 represents the structure where M1–M5 layers do not exist and only the M6 layer exists. It is the second combination, so it is numbered as 2 in 64-layer combinations. For a simplified model of a multi-layer structure with the same oxide layer property Xo= {to,αo,σo, Ey1} and the same metal
layer property Xm= {tm,αm,σm, Ey2}, the curvature 1/ρ of
the second combination R000001 under the six-metal CMOS MEMS process can be calculated as
−(tm(210 Ey2 σo− 210Ey1 σm)to2 +t2 m(390 Ey2 σo− 390 Ey1 σm)to + t3 m(180 Ey2 σo− 180 Ey1 σm))/
2401Ey12to4+ (868 Ey1 Ey2 + 7364 Ey12)tmto3 + (2382 Ey1 Ey2 + 8202 Ey12)t2
mto2
+(2188 Ey1 Ey2 + 3860 Ey12)t3
mto
+ (Ey22+ 670 Ey1 Ey2 + 625 Ey12)t4
m
.
Another example of the 63rd combination R111110 (all metals exist except M6) is (tm(210 Ey2 σo− 210Ey1 σm)to2 +t2 m(390 Ey2 σo− 390 Ey1 σm)to + t3 m(180 Ey2 σo− 180 Ey1 σm))/
2401Ey12to4+ (2660 Ey1 Ey2 + 5572 Ey12)tmto3 + (600 Ey22+ 5910 Ey1 Ey2 + 4074 Ey12)t2
mto2
+ (1200 Ey22+ 3980 Ey1 Ey2 + 868 Ey12)t3
mto
+ (625 Ey22+ 670 Ey1 Ey2 + Ey12)t4
m
.
Given the layer thickness and typical Young’s modulus of metal and oxide layers for the CMOS MEMS process, we may extract residual stress ofσmandσoby least-squares error minimization with beam curvature measurements of all or part of layer combinations with an offset. The residual stresses determine the slope and shape of the curvature versus layer combination curve and are independent of the curve offset in the simplified model. Therefore, residual stresses and offset can be extracted separately. Figure2shows the best curve fit of 32 measurements for the simplified model with parameters in table1.
Even numbers of layer combinations (32 combinations) are compared in figure2, since M6 is always present in our measurements as mentioned previously. The presence of each metal layer may have positive or negative contribution to the beam curvature, and the layer combination is binary coded. Therefore, the curve in figure2appears zig-zag and periodic
J. Micromech. Microeng. 23 (2013) 035023 F Y Kuo et al -150.00 -100.00 -50.00 0.00 50.00 100.00 150.00 200.00 250.00 300.00 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 1/ρ (1/Μ) Layer Combination Measurement Model Est. Error
Figure 2. Estimation error of the simplified model with measurement results.
-400.00 -300.00 -200.00 -100.00 0.00 100.00 200.00 300.00 400.00 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 1/ ρ ( 1/ Μ ) Layer Combination
Before Pack. (Meas.) After Pack. (Meas.) Before Pack (Model) After Pack (Model)
Figure 3. Estimation error of measurement data before and after packaging process.
in some sense. The average estimation error to curvature ratio is defined as the root square of the squared estimation error sum to squared curvature sum ratio. It is around 24% in this case with the offset value 185. From the analytical model, the other factor that determines the contribution of each layer to beam curvature is layer thickness. Therefore, the error of the simplified model can be further reduced by applying more accurate layer thickness of each layer in the CMOS MEMS process.
The complete analytical model can be imported into the electronic design automation (EDA) simulation tool using the Verilog-A format, and further incorporate with the curvature-to-capacitance model to derive a temperature-dependent sensing capacitance model of the cantilever beam with different oxide/metal combination for a given CMOS MEMS process. To model the temperature dependence of beam curvature at temperature T, the residual stress of any layerσ can be substituted by σ = σo+ α E T, where σois the residual stress at the reference temperature To.
3.3. Material yield due to high temperature packaging process
Figure 3 shows the parameter extraction result for measurement data before and after the high temperature packaging process for capping. Beam curvatures change significantly after the process. Based on the proposed model,
0.00E+00 5.00E+07 1.00E+08 1.50E+08 2.00E+08 2.50E+08 3.00E+08 0 100 200 300 400 Yield Stren g th ( P a) Temperature ( C)
Figure 4. Yield strength versus temperature for aluminum.
the parameters before and after the packaging process are extracted. From the analytical model, it is found that the term (σoEy2–σmEy1) in numerator dominates the trend of the curvature. The extracted parameters show that (σoEy2–σmEy1)
is about 2.03 × 1019before packaging and (σ
oEy2–σmEy1)
is increased to 2.66 × 1019 after the thermal process. The
change of the term is further analyzed.
The highest temperature for the CMOS MEMS packaging process is around 350 ◦C. During heating, the thermal expansion of composite materials introduces significant internal stress. At the same time, the yield strength drops at high temperature, as shown in figure 4 [12]. When the
-400 -300 -200 -100 0 100 200 300 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 1/ ρ ( 1/ Μ ) Layer Combination 20°C (Meas.) 20°C (Model) Est. Error -300 -200 -100 0 100 200 300 400 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 1/ ρ ( 1/ Μ ) Layer Combination 60°C (Meas.) 60°C (Model) Est. Error
Figure 5. Estimated curvature and estimation error for 20 and 60◦C with the proposed model.
Table 2. Extracted residual stress and curvature estimation error
after capping. 20◦C 40◦C 60◦C Est.σo 130 130 130 @ To(MPa) Est.σm {M1∼M5} : {−230, −240, −270, −300, −305} @ To(MPa) M6:−285 M6:−245 M6:−195
Est. MSE ratio 13.8% 14.4% 14.5%
of curvature (offset= 250)
stress reaches the yield strength of the material, especially metal layers, the material begins to deform plastically and releases the residual stress of metal layers. When the structure anneals and returns to normal temperature, the structure may accumulate more stress due to contraction. In section3.4, the thermal history of the packaging process will be analyzed in more detail.
3.4. Projection on curvature change due to thermal expansion Figure5shows the modeling error after parameter extraction for beam curvature after the high temperature packaging (capping) process, at 20, 40 and 60◦C with typical CTEαmand
αo applied to the residual stressσ= σo+ α E T. Table2 shows the estimated residual stresses and mean square error of estimated curvature values to actual measurements (MSE) with parameter extraction for 20, 40 and 60 ◦C. It is found
that the residual stress distribution of metal layers M1–M5 is not uniform as the simplified model in section3.2assumes. Table2 shows that residual stresses of M1–M5 are different but invariant with the temperature, while M6 changes with the temperature. In the following section, the distribution of residual stress due to metal yield is discussed.
3.5. Estimation of yield point based on stress distribution Based on the derived temperature dependency model, we observe the stress distribution of cross section for each layer of the composite beam structure. Figure 6(a) shows an example for stress distribution after the high temperature packaging process. The maximum stress –374 MPa appears near the bottom of the M1 layer for R100001. We further look into the temperature dependency of the maximum stress for each combination, and find that the maximum stress of all combinations is zero crossing at 190.2 ◦C. Figure 6(b) shows examples of R000101 and R100001. Considering that the yield stress of the aluminum drops rapidly around 200◦C, the result suggests that the common zero crossing temperature may indicate the point that most metal layers yield.
To further analyze the yield state of metal layers, we revisit the internal stressσbof metal layers in (5). When the metal starts to yield at high temperature, σb is constraint by the tensile yield strength Y of the metal
σb= min Y, (σ (z) + εE(z)) −E(z) ρ (z − zo) . (7)
J. Micromech. Microeng. 23 (2013) 035023 F Y Kuo et al
(a)
(b)
Figure 6. Stress distribution along z-axis cross section (where 0 of the x-axis represents the bottom side where M1 is located) and
temperature T (◦C) dependency of maximum stress for the combinations R000101 and R100001.
0 50 100 150 200 250 300 350 -3 -2 -1 0 1x 10 8 Temperature Internal Stress 0 50 100 150 200 250 300 350 -3 -2.5 -2 -1.5x 10 8 Temperature
Projected Residual Stress @ T0
0 50 100 150 200 250 300 350 -100 0 100 200 300 Temperature 1/rho M1 M6 M2 M3 M4 M5 M1 M6 M2 M3 M4 M5
Figure 7. Internal stress and projected residual stressσmof each metal layer at Towith tensile and compressive yields (R111111).
Table 3. Estimated residual stress and curvature with CTE scaling.
20◦C 40◦C 60◦C
Est.σo(MPa) 201 225 253
Est. Error ratio of curvature by yield analysis (K= 1.4) 14% 15% 23%
Est. Error ratio of curvature by residual stress extraction in section3.4 13.8% 14.4% 14.5%
Table 4. CAS tool output example for the simplified model in section3.2. R000001
−(tm(210 Ey2 σo− 210Ey1 σm)to2+ t
2
m(390 Ey2 σo− 390 Ey2 σm)to+ t
3
m(180 Ey2 σo− 180 Ey1 σm))
(2188 Ey1 Ey2 + 3860 Ey12)t3
mto+ (Ey22+ 670 Ey1 Ey2 + 625 Ey12)tm4)
(tm(210 Ey2 σo− 210Ey1 σm)t2
o+ t
2
m(390 Ey2 σo− 390 Ey1 σm)to + t3m(180 Ey2 σo− 180 Ey1 σm))/
(2401 Ey12t4
o+ (868 Ey1 Ey2 + 7364 Ey12) tmto3+ (2382 Ey1 Ey2 + 8202 Ey12)tm2t2o
+(2188 Ey1 Ey2 + 3860 Ey12)t3
mto+ (Ey2
2+ 670 Ey1 Ey2 + 625 Ey12)t4
m)
R000011
−(tm(336 Ey2 σo− 336Ey1 σm)to2+ tm2(624 Ey2 σo− 624 Ey1 σm)to+ tm3(288 Ey2 σo− 288 Ey1 σm))
(24 Ey22+ 3416 Ey1 Ey2 + 2608 Ey12)t3
mto+ (16 Ey2
2+ 1024 Ey1 Ey2 + 256 Ey12)t4
m)
(tm(336 Ey2 σo− 336Ey1 σm)t2
o+ tm(624 Ey2 σo2 − 624 Ey1 σm)to + t3m(288 Ey2 σo− 288 Ey1 σm))/
(2401 Ey12t4
o+ (1400 Ey1 Ey2
2+ 6832 Ey12) tmt3
o+ (12 Ey2
2+ 3780 Ey1 Ey2 + 6792 Ey12)t2
mt
2 o
+(24 Ey22+ 3416 Ey1 Ey2 + 2608 Ey12)t3
mto+ (16 Ey22+ 1024 Ey1 Ey2 + 256 Ey12)tm)4
R000101
−(tm(252 Ey2 σo− 252Ey1 σm)to2+ t
2
m(468 Ey2 σo− 468 Ey1 σm)to+ t
3
m(216 Ey2 σo− 216 Ey1 σm))
(96 Ey22+ 2816 Ey1 Ey2 + 3136 Ey12)t3
mto+ (52 Ey22+ 808 Ey1 Ey2 + 436 Ey12)tm4)
(tm(252 Ey2 σo− 252Ey1 σm)t2
o+ t
2
m(468 Ey2 σo− 468 Ey1 σm)to + t3m(216 Ey2 σo− 216 Ey1 σm))/
(2401 Ey12t4
o+ (1232 Ey1 Ey2 + 7000 Ey12) tmto3+ (48 Ey22+ 3228 Ey1 Ey2 + 7308 Ey12)tm2to2
+(96 Ey22+ 2816 Ey1 Ey2 + 3136 Ey12)t3
mto+ (52 Ey2
2+ 808 Ey1 Ey2 + 436 Ey12)t4
m)
R000111
−(tm(378 Ey2 σo− 378Ey1 σm)to2+ tm2(702 Ey2 σo− 702 Ey1 σm)to+ tm3(324 Ey2 σo− 324 Ey1 σm))
(144 Ey22+ 3996 Ey1 Ey2 + 1908 Ey12)t3
mto+ (81 Ey2
2+ 1134 Ey1 Ey2 + 81 Ey12)t4
m)
(tm(378 Ey2 σo− 378Ey1 σm)t2
o+ tm(702 Ey2 σo2 − 702 Ey1 σm)to + t3m(324 Ey2 σo− 324 Ey1 σm))/
(2401 Ey12t4
o+ (1764 Ey1 Ey2 + 6468 Ey1 2) tmt3
o + (72 Ey2
2+ 4602 Ey1 Ey2 + 5910 Ey12)t2
mt
2 o
+(144 Ey22+ 3996 Ey1 Ey2 + 1908 Ey12)t3
mto+ (81 Ey22+ 1134 Ey1 Ey2 + 81 Ey12)tm)4
R001001
−(tm(168 Ey2 σo− 168 Ey1 σm)t2o+ t
2
m(312 Ey2 σo− 312 Ey1 σm)to+ tm3(144 Ey2 σo− 144 Ey1 σm))
(216 Ey22+ 2576 Ey1 Ey2 + 3256 Ey12)t3
mto+ (112 Ey22+ 688 Ey1 Ey2 + 496 Ey12)tm4)
(tm(168 Ey2 σo− 168 Ey1 σm)to2+ t
2
m(312 Ey2 σo− 312 Ey1 σm)to+ tm(144 Ey2 σo3 − 144 Ey1 σm))/
(2401 Ey12t4
o+ (1232 Ey1 Ey2 + 7000 Ey12) tmto3+ (108 Ey22+ 3108 Ey1 Ey2 + 7368 Ey12)tm2to2
+(216 Ey22+ 2576 Ey1 Ey2 + 3256 Ey12)t3
mto+ (112 Ey2
2+ 688 Ey1 Ey2 + 496 Ey12)t4
m) 0 10 20 30 40 50 60 70 -400 -200 0 200 400 Combination 1/rho 0 10 20 30 40 50 60 70 -400 -200 0 200 400 Combination 1/rho model measurement model measurement
Figure 8. Beam curvature estimation with yield analysis (K= 1.4)
at 20◦C (top) and 60◦C (bottom).
The residual stresses σ (z) of metal layers are released when metal layers yield, so σ(z) is modified to σ(z) when yield
(σ(z) + εE(z)) −E(z)
ρ (z − zo) = Y for yield metal ⇒ σ(z) = Y + S
EE(z) + E(z)
ρ (z − zo).
(8)
σ(z) and ρ cannot be solved explicitly by (6) and (8) due
to the yield condition. However, we may deriveσ(z) andρ at T= T2 iteratively by (6) and (8) with an initial guess of ρ at T = T1near T2. With the iteration, state transition for a
given thermal history can be calculated. Figure7 shows the simulated scenario of the CMOS MEMS packaging process with the thermal history To → 350◦C→ 20◦C → 60◦C. The last cycle (20–60◦C) simulates the measurement activity (20–60◦C) in the lab. The simulation considers both tensile and compressive yield of metal layers.
It is found that the residual stress of each metal layer after the thermal process varies. This indicates that the assumption of single residual stress of metal layers used in previous sections is not valid. With the yield analysis, we derive the beam curvature for each layer combination without extracting the residual stress from measurement, as shown in figure8.
The estimation error of the model depends on the scaling factor K. With yield analysis in this section, similar
J. Micromech. Microeng. 23 (2013) 035023 F Y Kuo et al model accuracy without residual stress extraction is achieved
with typical CTE scaled by K = 1.4. The scaling factor K compensates inaccuracy of CTEs and the temperature-dependent yield curve. The yield analysis model does not require any pre-condition or extraction of residual stresses since the residual stresses can be predicted by simulating the yield/annealing process. Estimation results in table 3
simplified the analysis with the same yield stress curve for each layer. However, temperature-dependent yield stress of the aluminum thin film is process dependent and varies for different layer thicknesses [13]. Extracting residual stresses for individual layers, as shown in section3.4, is a more feasible approach for beam curvature estimation when detailed material yield data are not available.
4. CAS tool for analytical curvature modeling
Table4shows 10 Maxima CAS tool output examples of 26= 64 beam curvature formulas (R00001–R010011, the second to 20th combinations with M6 present) for the simplified model described in section3.2, where all oxide layers and all metal layers have the same thicknesses toand tm. Calculated formulas and extracted key parameters shown in section3can be used in the circuit simulator that provides CMOS MEMS designers accurate modeling of beam curvature based on the model extracted from the measurement results of silicon proven test structures.5. Conclusion
In this paper, we have demonstrated the analytical model of temperature-dependent yield effects on the curvatures of composite beam structures used in CMOS MEMS. Yield analysis during the thermal process of CMOS MEMS packaging process is also modeled. Key parameters including the residual stresses and the scaling factor of the CTE that characterize the temperature-dependent effect of the beam curvature of the process can be extracted by measuring curvatures of a limited number of metal/oxide layer combinations. The models are verified with measurement results of the ASIC-compatible 0.18μm 1P6M CMOS MEMS process before and after the high temperature packaging process. Beam curvature prediction in these models can be imported in EDA tools to model the temperature-dependent device characteristics such as sensing capacitance and spring constant of MEMS sensors for the monolithic sensor SOC design.
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