A Stress-Based Effective Film Technique for Wafer Warpage Prediction of Arbitrarily Patterned Films

Gregory T. Ostrowicki, Siva P. Gurrum

Texas Instruments, Inc.

12500 T I Blvd, Dallas, TX 75243

gtostrowicki@ti.com

Abstract

Initially flat silicon wafers are prone to warp due to the

high levels of intrinsic stress of deposited films, particularly

metallic films. Processing and handling of warped wafers in

the fab is a challenge. One of the ways to control the degree

of warpage is by limiting the amount of metallization allowed

on the wafer. However, this imposes a constraint on the

silicon designers, and can lead to decreased performance of

the IC. Therefore, there is a need to accurately predict the

amount of wafer warpage caused by a proposed layout in

order to give designers the most freedom to develop IC

solutions while ensuring that the processed wafers meet the

manufacturing equipment requirements.

The metal artwork (in addition to other materials, layer

thicknesses, processing parameters, etc.) is an important factor

in determining wafer curvature. Simple analytical methods,

such as Stoney’s Formula, cannot capture the non-uniform

warpage due to these patterned films. On the other hand,

numerical methods which require detailed modeling of the

film patterns across the whole wafer are computationally

expensive. Thus, a new finite element modeling technique

was developed in which the entire patterned film stack is

represented as a uniform effective orthotropic film bonded

onto a silicon substrate. The orthotropic properties are

determined from a small set of virtual experiments using a

unit-cell model that is characteristic of the actual pattern. The

resultant effective film, despite using a very course mesh, is

able to capture the non-uniform surface stress induced by a

patterned multi-layer film stack, and thus results in very

similar wafer warpage as in the conventional detailed model.

Several example film patterns will be presented here, where

the warpage difference between the detailed model and the

effective film model are less than a few percent across the

whole wafer.

1. Introduction

Wafers warp from the accumulation of stress within the

deposited films due to CTE mismatch, lattice mismatch,

impurities, recrystallization, creep, cure shrinkage, and other

phenomena [1]. It is a challenge to characterize all the

individual mechanisms that can contribute to the film stress

for each material in the stack. However, many of these

mechanisms can be lumped together into an “effective CTE”

(e.g. [2]). Therefore, this work focuses on thermal expansion

as the driving warpage mechanism.

As a silicon wafer goes through the IC manufacturing

flow, a series of patterned films are deposited across the wafer

to create a rectangular array of identical dies. Each die within

the wafer interior is essentially a unit-cell, with a

characteristic stress distribution through the film stack. Much

of the film stress is concentrated within the metal artwork, due

to the relatively high modulus and significant CTE mismatch

of metals with respect to the silicon substrate. Thus, the

patterning of the metal circuitry typically results in a non-

uniform biaxial stress depending on both the volume of metal

within the film as well as the design of the circuitry itself.

Any technique used to predict the global warpage of the wafer

after processing must therefore capture this non-uniform

stress in the film stack at the die level. For finite element

methods, this can require modeling the patterns exactly, using

element sizes that are on the order of the minimum feature

size in the film. Since the metal line width and thickness can

be in the micron to sub-micron scale, the necessary element

count can approach the millions for a single die depending on

the artwork, number of layers, and the die size. It can quickly

become computationally unreasonable to include this level of

detail across a whole wafer. For example, a 300 mm wafer

composed of 1x1 mm dies contains about 70,000 dies. To

simplify matters, one can model only a small portion of the

wafer and extrapolate the local curvature to estimate the

global warpage. However, this approach alone cannot capture

the effects of gravity, mechanical fixturing, and other global

body forces and constraints that can impact the warpage.

Previous numerical approaches have been used to model

global wafer warpage based on a film patterns of parallel lines

of a single material [3]. However, actual IC wafers can have

very complex film stacks. In this work, a finite element

approach is outlined to simplify the detailed nature of the

composite film stack into a uniform material with effective

properties, as shown in Figure 1. This representative film can

be modeled with a very coarse mesh, but still develop an

equivalent biaxial stress (and thus, equivalent warpage)

resulting from its CTE mismatch with the silicon substrate. A

detailed unit-cell model with representative film pattern

(typically one die) is first modeled in order to calculate the

effective film properties. Then, a second model of the wafer

with effective film can be used to determine the global

warpage by using a very coarse mesh.

Effective Film

Substrate

hf

Substrate

Composite Film Stack

hf

Figure 1. Simplification of composite film stackup into an

effective film with uniform properties.

978-1-4799-2407-3/14/$31.00 ©2014 IEEE 821 2014 Electronic Components & Technology Conference

2. Theory

Stress and Warpage

Stoney famously related the curvature of a plate to the film

stress through

( ) (2.1)

where R is the radius of curvature, E is the Young’s modulus,

h is the thickness, ν is the Poisson’s ratio, and the substripts f

and s correspond to the film and substrate, respectively [4].

This relationship holds for the case where the substrate and

film materials are uniform, isotropic, the film is much thinner

than the substrate, and the radius of curvature is much greater

than the substrate thickness (i.e. hf ≪ hs ≪ R). The film stack

on an IC wafer, however, is a mixture of different layers,

materials, and patterns. These heterogeneities can create a

non-uniform film stress, and thus non-uniform warpage along

the different radial directions of the wafer. This work aims to

simplify the composite film stack into a homogeneous

effective film for warpage prediction. Since any warpage

model must capture the directional nature of the film stress, an

anisotropic material model is used for the effective film.

Orthotropic Thin Film

A linear orthotropic material model is used to represent

the effective properties of an arbitrary composite film stack.

Since the film is thin and at a free surface, it can be assumed

to be in a state of plane stress, for which the conventions of

the stress components are shown in Figure 2.

σxx

σyyτxy σ11

σ22

θ=θp

Y

X Figure 2. Conventions for element in plane stress.

Hooke’s law for a thin orthotropic material can be

expressed in matrix form as

[

]

[

]

[

] (2.2)

where E is the Young’s modulus, G is the shear modulus, and

ν is Poisson’s ratio [5]. The following constraint is also

required to make the stiffness matrix symmetric:

(2.3)

In addition, there is an orientation θp at which the biaxial

stresses are aligned with the principal stresses and there is no

shear stress (i.e. τ = 0). The principal stresses σ11 and σ22 can

be determined by

( )

√(

)

(2.4)

and the principal orientation angle θp is determined by

(

) (2.5)

Therefore, for an orthotropic material in plane stress, and

oriented along the principal stress directions (i.e. θ = θp),

Hooke’s law can be further simplified to just the following

relationships

{

(2.6)

where the subscripts 1 and 2 refer to the principal directions,

and the corresponding stiffness symmetry constraint is

(2.7)

This result indicates that only the biaxial Young’s moduli

and Poisson’s ratios are required to fully define the stiffness

of the effective film, provided the film stress is oriented in the

principal stress directions. In addition, the film can be

assumed to have biaxial thermal expansion coefficients α1 and

α2. Thus for thermo-mechanical stress analysis, the effective

orthotropic film can be fully defined by only six material

parameters: E1, E2, v12, v21, α1, and α2.

Thermally Induced Stress-Based Effective Film

In the modeling approaches, a stress is induced in the film

by subjecting the wafer to a uniform change in temperature.

A 1-D approximation of such a bi-material strip undergoing

thermal loading is shown in Figure 3. Here the total strain of

each material must be equivalent as described by

(2.8)

where ΔT is the change in temperature from an initial stress-

free condition, and the subscripts f and s refer to film and

substrate, respectively. When the film is very thin (and

relatively soft) with respect to the substrate, the mechanical

strain in the substrate can be neglected (i.e. εs = 0) and (2.8)

can be expressed as

(2.9)

αf, Ef

αs, Es

Figure 3. 1-D illustration of bi-material strip under

thermal expansion

Although crystalline silicon is an anisotropic material, it

has been shown that a typical (100) oriented wafer has a

symmetric biaxial modulus [6], and its thermal expansion can

also be treated as isotropic [7]. Thus assuming an orthotropic

film on an isotropic substrate, (2.9) can be expanded along the

film’s principal directions to result in

822

{

(2.10)

where the subscripts 1 and 2 correspond to the film along the

respective principal directions. Substituting (2.6) into (2.10),

after some algebra results in

{

(2.11)

where

(

) (

)

(

) (

)

(2.12)

Equation (2.11) shows that for a given temperature

difference, the film’s principal stress is simply a linear

function of the substrate CTE, where the constants m and b

are purely functions of the film material properties. This

implies that effective orthotropic properties of a composite

film stack could potentially be determined if the average

thermally induced stress was measured for at least two

samples, on substrates with different CTEs. While this can be

impractical to exercise in a real experiment, it is

straightforward to implement in a numerical experiment. The

proposed process to determine the effective film properties is

thus outlined below:

1. Create detailed unit-cell model of the wafer with

representative film pattern (e.g. a single die).

2. Subject unit-cell to a characteristic temperature

change, and calculate the resulting volume-averaged

film stress along the principal directions.

3. Modify the CTE of the substrate in the unit-cell

model and repeat Step 2.

4. Solve for the constants m and b using (2.11) and the

results from Steps 2-3.

5. Find an appropriate property set for the film which

satisfies (2.7) and (2.12). The final step in the process outlined above presents a

potential caveat with this approach. As mentioned before, the

film is assumed to have six independent material parameters

(E1, E2, v12, v21, α1, α2). However, (2.7) and (2.12) provide for

only five constraint equations from which to solve for the six

properties. This in an unconstrained problem for which there

can be an infinite number of property sets which satisfy the

conditions. In the absence of a physical constraint, it is

possible to define an additional condition in an effort to obtain

a unique solution, such as assuming an isotropic CTE (i.e. α1

= α2). However, the authors feel that such a constraint may be

arbitrary, and in their experience may result in a poorly

defined material property set. Instead, an evolutionary

algorithm was used in EXCEL® to find a suitable material

property set, where the bounds of the effective modulus and

CTE were set to be within the maximum values of the

component materials in the film stack (i.e. 0 ≤ E1, E2 ≤ Emax

and 0 ≤ α1, α2 ≤ αmax), and the Poisson’s ratios were set to be

realistic values (i.e. 0 ≤ ν12, ν21 ≤ 0.5).

3. Numerical Models

A hypothetical wafer design was devised for use as a

baseline in order to demonstrate and compare the warpage

modeling approaches. A unit-cell of this wafer is shown in

Figure 4, and is representative of the patterning on a single

die. This die features alternating parallel strips of metal and

polymer film over a silicon substrate. Representative material

properties for the film and substrate are shown in Table 1.

Here hf = 10 um, hs = 725 um, b = d = 1 mm, wm = 71.4 um,

and vm = 0.5, where hf is the film thickness, hs is the substrate

thickness, b is the die width, d is the die depth, wm is the metal

line width, and vm is the volume fraction of metal in the film,

respectively.

For warpage modeling, only a small section of the wafer is

modeled due to element restrictions, but which is sufficient to

capture the wafer curvature in the absence of external forces.

As shown in Figure 5, the warpage model is a disk with radius

of 5 mm and is composed of a rectangular array of the

baseline die. Symmetry is utilized so that only ¼ of the model

is represented, where appropriate symmetry conditions are

imposed on the cut faces. Body forces such as gravity are

neglected such that the resulting curvature is purely a function

of the CTE mismatch between the film and substrate.

For the initial case, the entire film/substrate system is

assumed to be stress-free at elevated temperature and then

subjected to a drop in temperature to ambient conditions,

where ΔT = -100°C. In a more realistic scenario, however,

each individual film material is likely to have a unique stress-

free temperature depending on the material’s processing

history. This situation is addressed later as described in

Section 4, and is only briefly neglected here in order to

demonstrate the methodology of the approach. It should also

be noted that the film pattern was specifically chosen so that

the resulting principal stress directions in the film are aligned

with the coordinate system (i.e. σ11, σ22 = σxx, σyy). Die

designs where this is not the case are also discussed in Section

4.

Finite element models were developed using ANSYS®

14.5. First, a detailed model was constructed to establish a

benchmark of the wafer warpage. Then, a new approach was

pursued which first determined the effective properties of the

film based on stress within a unit-cell, and then applied the

effective properties to a simplified global warpage model.

823

silicon

metalpolymer

b

d

hs

hf

wm

Figure 4. Baseline die design with patterned film of

alternating metal and polymer strips on a silicon

substrate.

XY

Z

Figure 5. Baseline warpage model domain.

Table 1. Material properties of film and substrate.

Metal film Polymer film Si substrate

E 100 GPa 5 GPa 131 GPa

v 0.33 0.33 0.28

CTE 20 ppm/°C 40 ppm/°C 2.6 ppm/°C

Benchmark Approach: Detailed Model

A detailed finite element warpage model was constructed

using solid quadratic elements (SOLID186) with element

sizes of ~5x5x1 um in the film and ~5x5x145 um in the

substrate (see Figure 8). A sufficiently fine mesh was used to

capture the stress distribution through the film. The resulting

warpage is shown in Figure 6 for ΔT = -100°C. The radius of

curvature R was calculated along the X- and Y- axis by fitting

a circular arc to the out-of-plane displacement profiles shown

in Figure 7, which was found to be Rx = 11.492 m and Ry =

20.401 m. As a side note, if this curvature was extrapolated to

a 200 mm wafer, that would translate to a maximum warpage

of 435 um and 245 um along the X- and Y- axis, respectively.

X

Y

Uz (mm)

Figure 6. Benchmark warpage contours of detailed model.

-2.0E-04

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

0 1 2 3 4 5 6

Z-D

isp

lacem

en

t u

z(m

m)

Distance from center (mm)

X-Axis

Y-Axis

Figure 7. Warpage profile of benchmark model.

The disparity of the warpage along the two axes can be

explained by examining the stress within the composite film

as shown in Figure 8. Here the stress is concentrated in the

metal, and is much higher in the direction of the strips than

perpendicular to them. This is due to the fact that the metal is

less constrained near the sidewall adjacent to the low stress

polymer. In order to determine the effective film stress across

the wafer, the volume-averaged stress σ(avg)

in the film was

calculated to be σxx(avg)

= 123 MPa and σyy(avg)

= 90 MPa. It is

interesting to note that although the film stress in the X-

direction is only about 1.4X higher than in the Y-direction, the

curvature 1/R is almost 1.8X higher. Only a relatively small

non-uniformity in the biaxial stress is necessary to create a

significant difference in the warpage along the two

perpendicular axes.

824

Si

Polymer

Z

Y

PolymerMetal

σxx

σyy

Stress

(MPa)

Figure 8. Cross section of the detailed model near the

film/substrate interface. The corresponding contour plots

of σxx and σyy indicate that the overall film stress is more

tensile in the direction of the strips than perpendicular to

them.

New Approach: Stress-Based Effective Orthotropic Film

Whereas the benchmark approach involves using a

detailed model for the entire warpage domain, the new

approach employs a two-step process. First, a detailed model

is constructed only for one representative unit-cell (or die),

from which the smeared effective film properties are

calculated as described in Section 2. Then, a separate

simplified model, composed of just the substrate and effective

film, is used to determine the warpage over the entire domain.

Since the effective film has uniform properties across the

wafer, a relatively coarse mesh can be used for the warpage

model, which enables a much more efficient solution than the

benchmark approach.

The detailed unit-cell model was created for the die shown

in Figure 4, again using SOLID186 elements and a fine mesh.

Conventional unit-cell boundary conditions were applied on

the cut faces (i.e. out-of-plane displacements were coupled

across each of the four respective cut faces). The model was

subjected to ΔT = -100°C, and the resulting volume-averaged

biaxial film stress was determined. Then, a second identical

unit-cell model was made except that the CTE of the Si was

artificially changed to a different value, and again the

resulting volume-averaged biaxial stress was determined.

Although these two unit-cell models are sufficient to

determine m and b, a total of four models were created with Si

substrate CTEs of 1, 2.61, 5, and 10 ppm/°C in order to

clearly demonstrate that the trend is indeed linear as predicted

by (2.11) and shown in Figure 9. A suitable set of effective

film properties was found using the strategy described in

Section 2 with values listed in Table 1.

y = 67536x - 1.4403

y = 46347x - 1.0471

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

000.E+0 4.E-6 8.E-6 12.E-6

σf(a

vg) /Δ

T (

MP

a/

C)

αsubstrate (1/ C)

X-DirectionY-Direction

Figure 9. Biaxial film stress using detailed unit-cell model

as a function of substrate CTE. This data is from four

separate models in order to demonstrate that the trends

are linear. In general, two models are sufficient.

Table 2. Calculated effective film material property set.

Effective Film

Ex 56378 MPa

Ey 35973 MPa

νxy 0.2408

νyx 0.1536

αx 21.08E-6 °C-1

αy 22.96E-6 °C-1

Next, a warpage model was constructed using SOLID186

elements for the Si substrate and quadratic shell elements

(SHELL281) for the film. Unlike heterogeneous films which

can have a large variation in biaxial stress through the

thickness of the film (e.g. see Figure 8), the stress in a

homogeneous film is uniformly distributed through the film

thickness. Thus it is appropriate to use shell elements for the

effective film and this further reduces the model complexity.

Using the calculated effective film properties in Table 2, the

resulting warpage for this model is shown in Figure 10, and

can be seen to match very closely with the results of the

benchmark model in Figure 6. A comparison of the two

models is shown in Table 3, where the difference in the total

film stress and warpage is only 0-1.3%.

825

X

Y

Uz (mm)

Figure 10. Warpage contours of simplified model using

effective film properties.

Table 3. Comparison of baseline warpage models.

Detailed Model

Effective

Model % Difference

σxx(avg)

123.14 MPa 123.25 MPa 0.09%

σyy(avg)

90.27 MPa 91.05 MPa 0.86%

Rx 11.49 m 11.63 m 1.21%

Ry 20.40 m 20.13 m 1.32%

4. Additional Model Considerations

Arbitrary Patterns

For the example die design used in the previous analysis,

the principal stress in the film is naturally oriented with the

global coordinate system since the patterning is symmetric

with respect to the imposed unit-cell boundary conditions.

Therefore, it was sufficient to calculate the effective film

properties along the defined X- and Y- axes. However, for an

arbitrary die layout, the principal film stress may not be

aligned with the defined coordinate system. Since the

maximum warpage is typically in the direction of the

maximum principal stress, it is then necessary to rotate the

coordinate system so that the effective properties are

determined in the proper orientation.

To illustrate this procedure, consider the die layouts in

Figure 11. Design (a) is the same unit-cell as shown in Figure

4, with horizontally aligned strips where φ = 0°. Design (b)

has the same strip pattern except that the strips are at an angle

of φ = 45°. In fact, these two designs are really just

alternative unit-cells for the same repeating pattern of parallel

film strips. Therefore, both dies ought to have equivalent

principal stress magnitudes, and thus equivalent warpage.

The resulting volume-averaged film stress for each of these

unit-cell designs is shown in Figure 12, where each of the

three planar stress components were calculated in the rotated

X’-Y’ coordinate system, for 0°≤θ≤90°. Indeed, the principal

stresses for design (a) were found at θp(a)

= 0°, and are in close

agreement with those of design (b) where θp(b)

= 45°.

X′

YY′

θX

φ = 0° φ = 45°

φ

(a) (b)

Figure 11. Two alternative unit-cells that result in the

same global wafer design.

-40

-20

0

20

40

60

80

100

120

140

0 15 30 45 60 75 90

Fil

m S

tress

(M

Pa

)

θ (deg)

σxx' (φ=0°) σxx' (φ=45°)σyy' (φ=0°) σyy' (φ=45°)τxy' (φ=0°) τxy' (φ=45°)

Figure 12. Volume-averaged plane stress of the film in the

rotated X’Y’ coordinate system, where θ=0-90°. Two

cases are shown, for unit-cells (a) and (b).

The von Mises stress for the two designs is shown in

Figure 13. While the stress distribution is very similar for

both cases, there is some distortion near the cut boundary of

the die with angled strips. This is due to the fact that the

imposed unit-cell boundary conditions do not allow for global

shear deformation, and this enforces symmetry constraints

along the bounding faces even when the geometry is not

symmetric. Fortunately, this stress artifact is isolated to the

periphery of the die, and does not significantly affect the

volume-averaged stress calculations.

σvm

(MPa)

Figure 13. Von Mises film stress for the two unit-cell

designs. There is some artificial distortion in the stress

near the cut boundary of the angled strips due to the

asymmetric nature of the design with respect to the

imposed unit-cell constraints.

826

Multiple Stress-Free Temperatures

In all the previous models, the wafer was considered to be

stress-free at an initial temperature and then subjected to a

temperature ramp. In an actual wafer processing flow,

however, film materials are serially deposited, layer-by-layer,

at a wide range of temperatures and environmental conditions.

It is unlikely the wafer is stress-free at any given temperature.

In order to properly capture the component stresses of such a

heterogeneous film, the thermal assembly process can be

mimicked so that the appropriate film material is activated at

its respective stress-free temperature. This can be

accomplished using the element birth capability within the

finite element model.

Using the unit-cell from Figure 4 as an example, the

stress-free temperature Tref is assumed to be 120°C for the

metal film and 220°C for the polymer film, respectively. The

resulting stress distribution for the die at an ambient

temperature Tamb of 20°C can be determined using the strategy

shown in Figure 14, where the results of each load step from

the analysis are shown in Figure 15. In order to find the

effective film properties for this case, the model was repeated

using a modified Si substrate CTE as was done in Section 3.

Again the trends are clearly linear as shown in Figure 16,

from which appropriate effective film properties can be

calculated.

TrefA

Tamb

0 1

Birth (film A)

Birth (film B)

TrefB

2 3

Load Step

Tem

pera

ture

Figure 14. Example illustration of a multi-step film

deposition sequence, where each component film material

is activated through element birth at its respective stress-

free temperature.

Load Step 1, T=120°C

Load Step 2, T=220°C

Load Step 3, T=20°C

Figure 15. Progression of stress as the temperature is

ramped according to the film deposition sequence, and the

appropriate elements in the film are activated upon

reaching the stress-free temperatures of metal and

polymer, respectively.

y = 36722x - 0.8235

y = 28107x - 0.682

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

000.E+0 4.E-6 8.E-6 12.E-6

σf(a

vg) /Δ

T (

MP

a/

C)

αsubstrate (1/ C)

σx/ΔTσy/ΔT

X-Direction

Y-Direction

ΔT = -200°C

Figure 16. Biaxial film stress for unit-cell design with

different stress-free temperatures for metal and polymer

film.

5. Realistic Wafer

The methods outlined above were used to model the

warpage of a patterned wafer with a realistic die layout shown

in Figure 17. Here the metal artwork is partially covered by a

polymer film except for the circular regions that define an

exposed metal pad. The wafer thickness is 15 mil, and the

material properties used were the same as specified in Table

1, with stress-free temperatures of 120°C and 220°C for the

metal and polymer film, respectively.

827

Y

X

Figure 17. Example die layout, with metal artwork

covered by polymer film.

Both detailed and effective-film warpage models were

constructed for this wafer pattern in the same manner as

above, and subjected to the temperature history as shown in

Figure 14. The detailed model was composed of only a 4x6

die array to limit the computation expense, whereas the

effective film could be modeled for the entire wafer domain

due to its very low element count requirement. The results for

both models are compared in Table 4, where they are in

excellent agreement with a difference of just a few percent.

The principal stress orientation for this pattern resulted in the

maximum curvature to be in the Y-direction due to the pattern

of thin metal strips which run in parallel along the same

direction.

Table 4. Comparison of warpage models with realistic

pattern.

Detailed Model

Effective

Model % Difference

σxx(avg)

112.03 MPa 110.04 MPa 1.77%

σyy(avg)

118.85 MPa 116.98 MPa 1.58%

Rx 2.762 m 2.699 m 2.29%

Ry 2.399 m 2.421 m 0.91%

6. Conclusions

Wafer warpage is due not only to the amount of highly-

stressed metal content in the patterned film, but also the

design of the artwork. These patterns can have tendencies to

cause greater warpage in certain directions more so than

others, and cause handling issues during the manufacturing

flow. Conventional modeling of the warpage thus requires

modeling the fine details of the patterns, which is

computationally prohibitive for large areas of the wafer. A

better approach is to use an effective film with properties that

represent the cumulative effect of all the individual layers and

materials that make up the film stack on a patterned wafer.

A method for modeling the warpage of arbitrarily

patterned wafers was thus developed which uses an effective

orthotropic film to represent the composite film stack. A

detailed model of a representative unit-cell of the wafer was

first used to extract the material properties for an effective

film, based on thermal mismatch with the silicon substrate and

the resulting volume-averaged stress within the film stack.

Then, a global warpage model was constructed using the

effective film without having to capture the details of the

pattern. The stress-based effective film approach was also

demonstrated to be capable of capturing the unique stress-free

temperatures of each of the patterned film components.

The difference between the resulting biaxial warpage of

the simplified model with effective properties with that of the

detailed model was less than a few percent. Furthermore, the

proposed approach can be used to model the global warpage

across the entire wafer domain, which would be virtually

impossible to do with a detailed model. This approach

therefore allows for including the effects of external forces

such as gravity and mechanical fixturing which can impact the

wafer curvature.

References

1. P. A. Flinn et al., “Measurement and interpretation of

stress in aluminum-based metallization as a functino of

thermal history,” IEEE Trans. Electron Devices, vol. 34,

no. 3, pp. 689-699, Mar. 1987.

2. C.-H. Hsueh et al., “Residual stresses in thermal barrier

coatings: effects of interface asperity curvature/height

and oxide thickness,” Mater. Sci. Eng. A, vol. 283, pp.

46-55, 2000.

3. Y.-L. Shen et al., “Stresses, curvatures, and shape

changes arising from patterned lines on silicon wafers,” J.

Appl. Phys., vol. 80, no. 3, pp. 1388-1398, 1996.

4. G. G. Stoney, “The tension of metallic films deposited by

electrolysis,” Proc. R. Soc. Lond. A, vol. 82, no. 553, pp.

172-175, 1909.

5. N. E. Dowling, Mechanical Behavior of Materials 2nd

Ed., Prentice Hall, Upper Saddle River, 1999, pp. 190-

192.

6. M. A. Hopcroft, et al., “What is the Young’s modulus of

silicon?,” JMEMS, vol. 19, no. 2, pp. 229-238, Apr. 2010.

7. C. A. Swenson, “Recommended values for the thermal

expansivity of silicon from 0 to 1000 K,” J. Phys. Chem.

Ref. Data, vol. 12, no. 2, 1983.

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