effect of van der waals forces on reticle nonflatness in extreme

Effect of van der Waals forces on Reticle

Nonflatness in Extreme Ultraviolet Lithography

by

Harishanker Gajendran

Department of Civil and Environmental EngineeringDuke University

Date:

Approved:

Prof. Tod A. Laursen, Advisor

Dr. John Dolbow

Dr. Tom Witelski

Thesis submitted in partial fulfillment of the requirements for the degree ofMaster of Science in the Department of Civil and Environmental Engineering

in the Graduate School of Duke University2010

Abstract(Civil and Environmental Engineering)

Effect of van der Waals forces on Reticle Nonflatness in

Extreme Ultraviolet Lithography

by

Harishanker Gajendran

Department of Civil and Environmental EngineeringDuke University

Date:

Approved:

Prof. Tod A. Laursen, Advisor

Dr. John Dolbow

Dr. Tom Witelski

An abstract of a thesis submitted in partial fulfillment of the requirements forthe degree of Master of Science in the Department of Civil and Environmental

Engineeringin the Graduate School of Duke University

2010

Copyright c© 2010 by Harishanker GajendranAll rights reserved except the rights granted by the

Creative Commons Attribution-Noncommercial Licence

http://creativecommons.org/licenses/by-nc/3.0/us/

Abstract

Extreme ultraviolet lithography is recognized as the next generation lithographic

technique to meet the increasing demand of smaller and faster chips. Stringent

flatness requirements are enforced on the pattern surface to reduce pattern error

and to produce defect free chips. To meet this requirement, electrostatic chucking

is studied to explore the possibility to reduce the waviness of the chucked pattern

surface. Finite element analysis is done to predict the final reticle shape. A gap

dependent electrostatic pressure loading is considered. As the waviness is of the

order of nm, the mask-chuck interaction is modeled with van der Waals forces. The

analysis is run for 0-10000V force. A loading methodology was developed to model

the constraint free mask frontside during chucking. The peak to valley of the final

chucked surface obtained using penalty and van der Waals force method do not differ

substantially. The chucked frontside of the mask was found to be dependent on initial

mask and chuck waviness and the stiffness of the chuck.

iv

Contents

Abstract iv

List of Figures viii

Acknowledgements x

1 Introduction 1

1.1 Semiconductor Industry and its challenges . . . . . . . . . . . . . . . 1

1.2 Lithography Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Projection Lithography . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Immersion Lithography . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 Extreme Ultraviolet Lithography . . . . . . . . . . . . . . . . 4

1.2.4 Maskless Lithography . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.5 Nanoimprint Lithography(NIL) . . . . . . . . . . . . . . . . . 5

1.3 Objective of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Extreme Ultraviolet Lithography 8

2.1 EUVL set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Electrostatic Chucking . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Modeling non-flatness using B-splines . . . . . . . . . . . . . . . . . . 12

2.3 Finite element modeling details . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Mask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

v

2.3.2 Chuck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Contact Modeling of Electrostatic Chucking 15

3.1 Contact Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Contact Constraint Implementation . . . . . . . . . . . . . . . . . . . 21

3.2.1 Lagrange Multiplier . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.2 Penalty Method . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.3 van der Waals forces . . . . . . . . . . . . . . . . . . . . . . . 24

4 Effect of chucking on nonflatness - Numerical results 28

4.1 Validation of van der Waals force implementation in FEAP . . . . . 28

4.2 Effect of Chucking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.1 Nonflatness data 1 - Reticle 070425 . . . . . . . . . . . . . . . 32



5 Conclusion and Future Work 43

Bibliography 45

vi

List of Figures

1.1 SIA Roadmap [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 EUVL setup [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Bipolar Chuck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Mask/chuck - mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1 Two body contact problem [17] . . . . . . . . . . . . . . . . . . . . . 16

3.2 Contact traction vs gap [17] . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 L-J potential vs r/r0 [29] . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4 L-J potential aura of two bodies . . . . . . . . . . . . . . . . . . . . . 26

4.1 Modeling details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Comparison of exact vs FE solution . . . . . . . . . . . . . . . . . . 30

4.3 Mask Backside Nonflatness - Reticle 070425 . . . . . . . . . . . . . . 33

4.4 Mask Frontside Nonflatness - Reticle 070425 . . . . . . . . . . . . . . 34

4.5 Reticle 070425 Profile at x=0 crosssection . . . . . . . . . . . . . . . 35

4.6 P-V vs Voltage variation for Reticle 070425 . . . . . . . . . . . . . . . 35

4.7 Mask Backside - Nonflatness Reticle 070622 . . . . . . . . . . . . . . 37

4.8 Mask Frontside - Nonflatness Reticle 070622 . . . . . . . . . . . . . . 38



4.11 Mask Backside - Nonflatness Reticle 080408 . . . . . . . . . . . . . . 40

4.12 Mask Frontside - Nonflatness Reticle 080408 . . . . . . . . . . . . . . 41

vii



viii

Acknowledgements

I am grateful to Prof.Tod Laursen for his invaluable guidance and patience with me.

My association with him has been a great learning experience over the past three

years. I also wish to thank my committee members, Dr.John Dolbow and Dr.Tom

Witelski for their participation and expertise.

I wish to acknowledge Prof.Sanjay Govindjee and University of California, Berke-

ley subcontract from Intel Corporation for providing me with an opportunity to work

on this project.

My special thanks are due to Anand, Chandu, Frederick and Vindhya for their

timely help and suggestions. I also wish to thank my friends Justin, Jessica Sanders,

Jessica Berry, Temesgen, Terence and Wen for the informative journal club meetings

and fun discussions.

ix

1

Introduction

1.1 Semiconductor Industry and its challenges

Integrated circuit technology provides us the comfort of the electronic devices such as

iphones, mp3 players, laptops at an affordable price and in compact sizes. According

to Moore’s law, the number of devices on a chip doubles every 18 months [23].

The semiconductor industry has been able to achieve this steady increase in the

integration density of electronic components on a silicon chip for over three decades.

The success of the semiconductor industry lies in its ability to keep the production

cost of printing a silicon wafer constant while increasing the number of transistors per

wafer. The driving technology for this tremendous increase in integration density has

been optical lithography. In optical lithography, the image on the mask is projected

onto the silicon wafer with a factor of reduction 4:1 using an optical projection tool

[13]. Optical lithography throughput has been improved over decades by introducing

technological enhancements in lens and imaging material technology and light sources

with smaller wavelengths. The current rate of miniaturization can be sustained by

improving current optical projection lithography techniques while simultaneously

1

Figure 1.1: SIA Roadmap [14]

developing next generation lithographic technologies as the cost of pushing optical

lithography will be beyond the conceivable limit of introducing new technology.

The important parameter to be considered in conventional optical lithography

are resolution limit and depth of focus, which are given by Raleigh’s equation [13],

R = k1λ

NA(1.1)

DOF = k2λ

NA2(1.2)

where λ is the exposure wavelength, NA is the numerical aperture and k1, k2 are

constants that depend on the specific resist material, process technology and image-

formation technique.

Device miniaturization(high resolution) can be obtained by using shorter wave-

length light and high numerical aperture lenses. In such high numerical aperture

2

systems, the depth of focus becomes very small( inversely proportional to the square

of NA) and hence, the exposure process becomes sensitive to slight variations in the

thickness and absolute position of the resist layer.

The miniaturization of the circuit results in improvement of the chip performance

and they have gone hand in hand with increase in chip yield and reduction in cost.

The critical size of the component has decreased from 15 µm three decades ago to 180

nm. But the manufacturing technology has remained the same, optical lithography

in a qualitative sense. To sustain this phenomenal rate of increase in chip density,

Semiconductor Industry Association(SIA) provides a roadmap(Fig.1.1) for the future

technology and a timeline to develop next generation technology to meet consumer

demand in the semiconductor industry.

1.2 Lithography Techniques

This section provides an introduction to various lithographic techniques.

1.2.1 Projection Lithography

In 1960, projection lithography was found ideal for printing IC’s. The basic principle

of projection lithography is the projection of pattern on the mask onto the wafer

using exposure tools. A quasi-monochromatic spatially incoherent light source is

used to illuminate the mask. The source light is homogenized to ensure high uniform

density distribution at the plane of mask. The light transmitted through the mask is

collected by a projection lens which images the mask onto the wafer with a reduction

in magnification(4:1). To improve resolution, the wavelength has steadily decreased

from 365 nm (i-line of mercury) to 257 nm (high pressure mercury arc lamp) to

248 nm(KrF laser) and presently at 193 nm(ArF excimer laser) [13]. The resolution

and depth of focus can be further improved through phase shift masks, modified

illumination systems and greater flatness of the wafer.

3

1.2.2 Immersion Lithography

Immersion lithography [24] is a resolution enhancement photolithographic technique

to produce smaller features with 193 nm wavelength light. The basic principle is to

increase the refractive index of the medium between the lens and wafer surface by

replacing air with water. Current immersion lithography can achieve feature sizes

below 45 nm. The major obstacles of immersion lithography system are defect control

due to particle impurities, bubbles in fluid, temperature and pressure variations in

fluid and fluid absorption by resist. 193 nm light can ionize the water producing

solvated electrons which can affect the resolution performance. High index fluids

such as highly fluorinated polyether, hydrocarbons, high index optical materials are

probable candidates to produce 45 nm node and below. Due to the rising process

and patterning complexity, exploration of other technology is needed for 22 nm node

and below.

1.2.3 Extreme Ultraviolet Lithography

Extreme Ultraviolet Lithography(EUVL) [4] is a logical extension of optical lithogra-

phy with 13.5 nm extreme ultraviolet wavelength light to produce 22 nm node or less.

EUVL follows Rayleigh’s equations and hence provides better resolution and depth of

focus. But at the extreme ultraviolet regime, all materials are highly absorbing and

hence, an all-reflective optic system is used instead of a lens based refractive system.

A detailed explanation of EUVL is given in the next chapter. The challenges faced

by EUVL are the production of masks with low defect density, EUV source with

high output power, highly reflective and less absorbing mirror systems, protection

of masks from contaminants and the production of masks according to Semiconduc-

tor Equipment and Materials International(SEMI) standards to minimize placement

errors.

4

1.2.4 Maskless Lithography

In maskless lithography [24], the difficulty, time and cost associated with designing

and repairing the mask is circumvented by focusing the radiation to a narrow beam

instead of projecting through or transmitting through a photomask. The focused

beam is used to write the image on the wafer directly, thus eliminating the mask.

The main advantage of ML is high resolution and it works with numerous resist

materials. The drawbacks are that the ML tools are slow and expensive.

1.2.5 Nanoimprint Lithography(NIL)

In NIL [1], a mold with nanostructure on its surface is pressed onto a thin resist of

the wafer, which creates a thickness contrast pattern in the resist. An etching process

such as reactive ion etching(RIE) is used to remove the residual resist in the com-

pressed area. NIL has the capability to produce 32 and 22 nm nodes. The advantage

of NIL is low cost, high throughput and high resolution technology. NIL is still at its

infancy and further investigations are needed to make it a manufacturing technology.

The primary issues with NIL are the surface sticking problem(release of the imprint),

particle contamination and the effect of thermal expansion on lithography resolution.

1.3 Objective of thesis

Due to the increasing cost of the enhancement techniques in projection lithography

and the required time in developing the new technology as a feasible manufactur-

ing process, EUVL is considered as the leading candidate for production of 22 nm

node and less. In EUVL, the mask is held electrostatically against the chuck. This

electrostatic chucking process affects the nonflatness of the mask due to the contact

interaction and the electrostatic force between the mask and chuck. A fundamental

understanding of the chucking phenomenon is required to realize the SEMI P37 and

5

SEMI P40 stringent flatness requirements. Finite element analysis of electrostatic

chucking to identify the shapes that can be flattened was done by Mikkelson et al.

[20]. In this study, Legendre polynomials were used to model the waviness of the

mask chuck surface. In the final chucked shapes, lower order modes of nonflatness was

found to be flattened. In this analysis, the electrostatic pressure was assumed to be

independent of gap. Mikkelson et al. [21] showed that the thickness non-uniformity

has a dominant effect on the flatness of the chucked mask frontside. The simulations

were run for two different mask materials with friction effects considered. The effect

of chuck stiffness on the waviness of the mask frontside after chucking was studied in

[22]. The peak-valley of the mask and chuck surfaces were found to be dependent on

low clamping pressures and was insensitive to furthur increase in pressure. Nataraju

et al. [25] conducted electrostatic chucking experiments in vacuum and observed a

good comparison between the finite element results and experimental data. Both

flat chuck and pin chuck were considered. In all the above mentioned work, impene-

trability condition was enforced using the penalty method. The electrostatic loading

was considered independent of gap between the mask and chuck.

The primary challenge is to understand and characterize the ability of electro-

static chucking phenomenon to acheive consistent and reliable shapes of chucked

masks. The objective of this thesis is to study the effect of initial nonflatness of

mask and chuck, chucking voltage, chuck and mask dimension and gravity on the

final nonflatness of the mask. A finite element model of the mask and chuck with

initial nonflat surface using B-splines is developed. A gap dependent electrostatic

loading is considered. To predict the final nonflatness of the mask frontside with

nm accuracy, the contact interaction between mask and chuck is modeled using van

der Waals forces. These results are compared with final waviness obtained using the

penalty method. The major contributions of this work are:

6

1. Considering the effect of gap dependent electrostatic loading

2. Electrostatic chucking analysis with van der Waals forces

1.4 Outline of thesis

An introduction to Extreme ultraviolet lithography is given in chapter 2. The model-

ing details of the nonflatness using B-splines and a brief description of finite element

model of the mask and chuck is also given. Chapter 3 provides a detailed descrip-

tion on the implementation of contact constraints using penalty and van der Waals

force method. Final chucked shapes and peak-valley comparison of the waviness be-

tween penalty and van der Waals forces are presented in chapter 4. Summary and

conclusion are presented in chapter 5.

7

2

Extreme Ultraviolet Lithography

2.1 EUVL set up

As optical lithography technology is approaching its resolution limit, EUVL is con-

sidered to be the next generation lithography technology for 22 nm node and below.

Current lithographic techniques uses light in deep ultraviolet range at about 248 nm

wavelength to print 150-120 nm size features on chips. EUVL systems uses extreme

ultraviolet light with wavelength of 13.5 nm to print features ≤22 nm. EUVL follows

the Rayleigh’s equation to control resolution and DOF. As the exposure wavelength

are less than one tenth of wavelength in current lithographic techniques, conventional

exposure system will no longer be suitable.

Fig.2.1 shows the schematic of EUVL system [16]. EUV radiation with wave-

length in the order of 13.5 nm is obtained from laser produced plasmas. A laser is

directed into a source element such as xenon, tin or lithium which decay these ions

to produce EUV radiation [3]. There are 13 reflective surfaces in a EUV exposure

system, 6 mirrors in illumination optics, 1 mask, 6 mirrors for projection optics [26].

With reflectivity of each mirror being 70%, for a throughput of 100 wafers/hour, a

8

Figure 2.1: EUVL setup [16]

light source of 115 W is required. Laser produced plasma(LPP) sources are debris

free which increases the life time of the collector mirror. Condensor optics collects

and shapes these radiations. As EUV radiation is strongly absorbed in all materials,

the use of refractive optical elements such as lenses has to be replaced by reflec-

tive imaging systems. The reflective surfaces are coated with multilayer thin films

which consist of a large number of alternating layers of materials having dissimilar

EUV optical constants. The collective coherent radiation is then focused onto the

reflective mask. The mask frontside contains the pattern which is reflected onto the

wafer using a 4:1 reflective reduction optic system. Conventional lithographic imag-

9

ing systems using wavelengths of 193 nm and above are telecentric on both reticle

and wafer space. The magnification of such systems are invariant with respect to

object or image deviation from their respective focal planes. EUV imaging systems

are non-telecentric, i.e the reticle is illuminated by rays that are 6 degrees off axis.

Out of plane distortions causes points on the reticle to be out of the optimal focal

plane and therefore resulting in magnification and registration errors. The lack of

telecentricity on the object side places a severe requirement on reticle flatness. SEMI

P37 specifies that nonflatness of reticle surface to be no more than 30-100 nm P-V.

SEMI P40 requires 50 nm nonflatness of chucking surface [25].

2.1.1 Electrostatic Chucking

Electrostatic chucking [2] is used to hold the mask against the chuck to avoid scratches

and for safe handling of the mask. Consider the mask chuck system as shown in

Fig.2.2(a). The two metal electrodes inside the chuck are connected to a voltage

source. The positively charged electrode creates a negative electric field on one

side of the mask as shown in Fig.2.2(b) and vice versa. The dielectric coating on

the chuck and the gap between the mask backside and chuck frontside acts as an

insulator. Thus, the mask chuck system behave like a parallel plate capacitor and

with the application of a voltage V between the two plates, an attractive electric

force is generated and applied to the reticle. These chucks are called bipolar chucks

[24].

The mask-chuck capacitor system is separated by a dielectric film and a vacuum

gap due to the nonflatness of the mask-chuck. Thus, the capacitance of the overall

system can be derived using capacitance in series equation. The capacitance due to

the gap and dielectric film is given respectively as [9],

C1 =ε0A

gC2 = k

ε0A

d(2.1)

10

(a) Schematic

45

Substituting Eq. (3.4) in the above equation, the expression for the electrostatic force F is

δddC

VF2

2

1−= . (3.6)

The pressure, P, due to the electrostatic force, F, is

2

0

22

)(2 dK

KV

A

FP

+== δ

ε. (3.7)

Note that the above derivation directly applies to the case of an ESC having a single

embedded electrode and the wafer or mask serving as the other electrode. This is the

monopolar chuck configuration, which is discussed in Section 3.3.

3.3 Monopolar and Bipolar Chucks

Electrostatic chucks can be classified on the basis of the number of electrodes.

The most common are chucks with one (monopolar) and two (bipolar) electrodes. Figure

3.2 shows the basic concepts of the two designs.

(a) (b)

Fig. 3.2. (a) Monopolar and (b) bipolar chuck designs [3.13].

In the monopolar case, the wafer (or mask) has to be electrically connected to

form one of the electrodes. The wafer potential is therefore well defined and there is no

charging. However, making electrical contact to the mask is difficult and may lead to

particle generation [3.6].

(b) Principle of OperationFigure 2.2: Bipolar Chuck

where ε0 is the permittivity of free space, k is the relative permittivity of the dielectric

material, A is the surface area of the mask, g is the gap due to nonflatness of

mask/chuck system and d is the dielectric film thickness. The equivalent capacitance

of the system is,

C =C1C2

C1 + C2

(2.2)

which reduces to

C =kε0A

d+ kg(2.3)

The stored energy of the capacitor is given as,

U =1

2CV 2 =

1

2

Q2

C(2.4)

where V is the applied voltage and Q is the stored charge. The electrostatic pressure

generated can be derived from the stored energy of the capacitor as

P =F

A=

1

A

(∂U

∂g

)Q

(2.5)

Using eqs. (2.1) - (2.4), the electrostatic pressure can be written as

P =F

A=

V 2k2ε0

2 (kg + d)2 (2.6)

11

In monopolar chucks, the need for electrical contact in the mask might require special

time for clamping and might generate particles. To prevent any damage to the mask,

bipolar chucks are used. In bipolar chucks, two oppositely charged electrodes are

placed inside the chuck with a closed circuit, thus avoiding any electrical contact with

the mask. As the voltage V is split between the electrodes in the chuck, V = V2

. In

this thesis, bipolar chucks are considered for the simulation.The electrostatic pressure

of this system is given as

P =V 2k2ε0

8 (kδ + d)2 (2.7)

2.2 Modeling non-flatness using B-splines

The initial nonflatness of the mask and chuck is measured using an interferometer,

an optical measurement technique of the surface waviness. The interfermoteric data

[6] consists of x,y locations of the surface with corresponding height, z data. This

discrete data is modeled with B-splines to produce a continuous surface. Modeling

of nonflatness using B-splines used in this thesis, is a work done as a part of master

thesis, by Gerd Brandsetter at UCB [5]. A brief explanation of the modeling details

of nonflatness using B-splines is provided for completeness. Given a set of points, a

curve representation of the form

C(x) =n∑i=0

Bi(x)Pi (2.8)

is desired, where Pi are control points and Bi(x), i = 0, 1, . . . , n are piecewise poly-

nomial functions. A recursive algorithm is used to generate B-splines of higher order

from B-splines of lower order. Given a knot vector,

t0, t1, t2, t3, t4, t5, . . . tn, tn+1, tn+2, tn+3, tn+4 =

x0, x0, x0, x0, x2, x3, . . . xn−2, xn−1, xn, xn, xn (2.9)

12

B-splines of order 0 are given as,

Bi,0(x) =

1 if ti ≤ x ≤ ti+1

0 otherwise

, t = 0, 1, . . . n+ 3 (2.10)

B-splines of higher order are derived as,

Bi,k(x) =x− titi+k − ti

Bi,k−1(x) +ti+k+1 − xti+k+1 − ti+1

Bi+1,k−1(x) (2.11)

Bi,k(x) ≡ 0 if ti = ti+1 (2.12)

Cubic B-splines are used to model the waviness. The surface can be modeled using

B-splines in 2D, along x and y, as

S(x, y) =n∑i=1

l∑j=1

ci,j Bi,3(x)Bj,3(y) (2.13)

For the given zygo data,

(xj, yj, zj) j = 1, 2, . . . m (2.14)

where, m is the number of data points, a least square fit is used to determine the

coefficients, ci,j.

m∑j=1

(S (xj, yj)− zj)2 → min (2.15)

2.3 Finite element modeling details

In this section, the finite element modeling details of the chuck and mask are de-

scribed.

2.3.1 Mask

A mask of size 140.78 × 140.78 × 6.5 mm is modeled and meshed with solid elements.

The mask frontside and backside is warped using the spline coefficients calculated

13

Figure 2.3: Mask/chuck - mesh

for the given zygo measurement data. In this work, three different nonflatness data

are analyzed. The mask is modeled with material properties of an ULE glass with

an Young’s Modulus of 30 GPa, a Poisson’s ratio of 0.2 and density of 2.31 g/cm3.

2.3.2 Chuck

A chuck of size 140.78×140.78×22.6 mm is modeled with solid element as shown

in Fig.2.3. The chuck is modeled with ceramic material properties, with Young’s

modulus of 380 GPa and a Poisson’s ratio of 0.2. The chucked front side is warped

using a warp function for the given nonflatness data. Gravity is included in the

analysis. The chuck backside surface is held fixed at three points. The loading

methodology to simulate the chucking phenomenon is explained in detail in chapter

4.

14

3

Contact Modeling of Electrostatic Chucking

In this chapter, a general contact formulation of a kinematically linear elastic prob-

lem is presented [17]. The strong and weak form of a two body contact problem

is described. The impenetrability condition of a contact surface pair is modeled

through a contact functional. The contact functional for the Lagrange multiplier

method, penalty method and van der Waals force method are elaborated. Friction-

less contact is considered throughout this chapter for the simplicity of description of

the formulation. In the simulation of the chucking of the real mask-chuck system,

friction is included.

3.1 Contact Formulation

Consider two bodies Ω(1), Ω(2) as shown in Fig.3.1, whose boundaries consists of three

regions Γci, Γσ

i, Γui, such that

Γσ(i) ∪ Γu

(i) ∪ Γc(i) = ∂Ω(i) (3.1)

Γσ(i) ∩ Γu

(i) = Γσ(i) ∩ Γc

(i) = Γu(i) ∩ Γc

(i) = φ (3.2)

where, Γu(i) is the region where displacement are specified, Γσ

(i) represents the region

15

Figure 3.1: Two body contact problem [17]

where traction are prescribed, Γc(i) is the expected contact surface. As the main focus

of this chapter is to provide the formulation details of the methods to implement the

contact constraint, a kinematically linear elastic problem is considered. For a given

point x ∈ Γc(1), the closest point on Γc

(2) is found from the relation,

y(x) = arg miny∈Γ

(2)c‖x− y‖ (3.3)

A gap function g(x) is defined using the contact point pair obtained from eq.(3.3)

as,

g(x) = −[x + u(1)(x)− y(x)− u(2) (y(x))

]· ν(x), ∀ x ∈ Γc

(1) (3.4)

where, u(1)(x) and u(2)(y(x)) are displacements of corresponding contact pair, ν(x)

is the outward normal to Γc(2) at point y(x). The above equation can be rewritten

as

g(x) = g0(x)−(u(1)(x)− u(2) (y(x))

)· ν(x) (3.5)

where, g0(x) is the initial gap, which is given as,

g0(x) = − [x− y(x)] · ν(x) (3.6)

16

The impenetrability condition is enforced by requiring the gap function to satisfy

g(x) ≤ 0 (3.7)

where, positive gap indicates interpenetration and a negative value indicates an open

gap. Let t(i) denote the traction acting on the contact surface Γc(i). According to

Newton’s third law, the tractions t(1) and t(2) must be equal and opposite, i.e

t(1)(x) = −t(2) (y(x)) ∀x ∈ Γc(1) (3.8)

As the contact traction at Γc can now be expressed as a function of either t1 or t2,

the contact normal pressure tN acting on x can be given as,

tN(x) = t(1)(x) · ν(x) = ν(x) · σ(1)(x) n(1)(x) (3.9)

where, σ(1) is the Cauchy stress and n(1) is the outward normal at x on Γc1. As

the normal at the contact pair doesnt depend on the displacement field for the

kinematically linear elastic problem, ν(x) = −n(1)(x). Hence, eq.(3.9),(3.5) and (3.6)

can be rewritten as,

tN(x) = −n(1)(x) · σ(x)n(1)(x) (3.10)

g(x) = g0(x) +(u(1)(x)− u(2) (y(x))

)· n(1)(x) (3.11)

g0(x) = (x− y(x)) · n(1)(x) (3.12)

To introduce friction into the model, consider the tangential motion of a point

x ∈ Γc1 relative to the opposing surface Γc

2, which is given as,

uT(x) = u(1)(x)− u(2) (y(x))− [(u(1)(x)− u(2) (y(x)) · ν]ν (3.13)

= [I− ν ⊗ ν](u(1)(x)− u(2) (y(x))) (3.14)

Similarly, the tangential tractions acting at x ∈ Γc1 can be obtained by resolving

the traction t(1)(x) as,

tT(x) = −[t(1)(x)− (t(x)(1) · ν)ν] = − [I− ν ⊗ ν]t(1) (3.15)

17

In this work, a Coulomb frictional law is used to model the friction. With the above

definition of the gap and contact tractions, the strong form of the linear elastic

frictional contact problem can be summarized as:

To find the displacement field u(i) : Ω(i) × I→ <nsd such that,

1. Linear Momentum balance in Ω(i), i = 1,2

σ(i)kj,j + fk = 0 (3.16)

2. Boundary conditions

σ(i)kj n

(i)j = t

(i)k in Γ(i)

σ (3.17)

u(i)k = u

(i)k in Γ(i)

u (3.18)

3. Strain displacement relation

ε(i)kj = u

(i)(k,j) =

1

2

(u

(i)k,j + u

(i)j,k

)(3.19)

4. Constitutive relation, linear elasticity

σ(i)mn = C

(i)mnkl ε

(i)kl (3.20)

5. Contact conditions ∀x ∈ Γ(1)c

tN ≥ 0

g ≤ 0

tN g = 0

φ(tT, tN) = ‖tT‖ − µtN ≤ 0 (3.21)

uT = γtT

‖tT‖

γ ≥ 0

γφ = 0

18

where, φ is the slip function which restricts the tangential traction to not to exceed

the coefficient of friction times the contact pressure tN .

The above strong from of the two body contact problem can be transformed to

an equivalent weak form in a variational context. For the sake of brevity and better

representation of van der Waals forces, friction is neglected in the formulation until

end of this chapter. Let the solution space U i and the weighting space V i be defined

as,

U (i) =

u(i) : Ω(i) → Rnsd |u(i) ∈ H1

(Ω(i)),u(i) = u(i) in Γ(i)

u

(3.22)

V(i) =

w(i) : Ω(i) → Rnsd |w(i) ∈ H1

(Ω(i)),w(i) = 0 in Γ(i)

u

(3.23)

where, H1(Ω(i)) denotes a set of square integrable derivative function on Ω(i). The

weak form can be derived by weighting the residual of the momentum balance equa-

tion(eq.(3.3)) with an arbitrary function w(i) ∈ V i in an integral sense, which is given

as,

G(i)(u(i),w(i)

)=

∫Ω(i)

(ρ(i) w

(i)j u

(i)j + w

(i)j,k σ

(i)jk

)dΩ(i)

−∫

Ω(i)

w(i)j fj dΩ(i) −

∫Γ

(i)σ

w(i)j t

(i)j dΓ(i) (3.24)

−∫

Γ(i)C

w(i)j t

(i)j dΓ(i) = 0 ∀ wi ∈ V i

Let U be the collective set of U (i) and V be the collection of V(i). Then the eq.(3.24)

19

for i = 1, 2 can be added together and rewritten as,

G(u,w) =2∑i=1

G(i)(u(i),w(i)

)

=2∑i=1

∫Ω(i)

(ρ(i) w

(i)j u

(i)j + w

(i)j,k σ

(i)jk

)dΩ(i)

−∫

Ω(i)

w(i)j fj dΩ(i) −

∫Γ

(i)σ

w(i)j t

(i)j dΓ(i) (3.25)

−∫

Γ(i)C

w(i)j t

(i)j dΓ(i)

= 0

where, the last integral over Γ(i)c is the contact virtual work, Gc(u,w). The total

virtual work is the summation of internal/external virtual work and the contact

virtual work.

G(u,w) = Gint,ext(u,w) +Gc(u,w) (3.26)

Using Newton’s law, the contact virtual work can be rewritten as,

Gc(u,w) = −∫

ΓC(1)

(w(1)(x)−w(2) (y(x))

)· t(1)(x) dΓ(1) (3.27)

Eq.(3.25) can also be obtained as a solution to a constrained minimization problem

of total energy Π of the system. The total potential energy associated with the two

body problem is given as,

Π(u, λN) =2∑i=1

Πi(ui) + Πc(u, λN) (3.28)

The energy associated with each body due to internal and external forces can be

given as,

Π(i)(u(i))

=

∫Ω(i)

W (i) dΩ −∫

Ω(i)

f(i) · u(i) dΩ −∫

Γ(i)σ

t(i) · u(i) dΓ (3.29)

20

where, W i(ε(i)) is the stored elastic energy which is expressed as,

W(i)(ε(i)) =1

2ε(i):Ci:ε(i) (3.30)

where, Ci is the elastic constant tensor. Consider the potential energy associated

with each body Ωi generated with a displacement field u(i) + αw(i), where, α is a

scalar,

Π(i)(u(i) + αw(i)) =

∫Ω(i)

W (i)(ε(i)(u(i) + αw(i))

)dΩi −

∫Ω(i)

f(i).(u(i) + αw(i)) dΩi

−∫

Γσ(i)

t(i).(u(i) + αw(i)) dΓi (3.31)

Equilibrium configuration is obtained by taking the directional derivative of the

eq.(3.31). The directional derivative is taken by differentiating the above equation

w.r.t α,

d

dα

∣∣∣∣α=0

Π(i)(u(i) + αw(i)

)= 0 (3.32)

which can be simplified to,∫Ω(i)

σ(i) : [∇w(i)] dΩi −∫

Ω(i)

f(i).w(i) dΩi −∫

Γσ(i)

t(i).w(i) dΓi = G(i)(u(i),w(i))

(3.33)

Eq.(3.33) obtained through minimization of the potential energy is the same as

Gint,ext in eq.(3.25). This equivalence between the weighted residual method and

minimization of the potential energy is shown, so that the contact potential Πc can

be used in describing the following methods to enforce the contact constraints.

3.2 Contact Constraint Implementation

In this section, a general description of the methods to enforce the contact constraint

are described.

21

3.2.1 Lagrange Multiplier

Lagrange multiplier problem is obtained by considering the following contact func-

tional,

Πc =

∫Γic

λNg dΓi (3.34)

where λN is the Lagrange parameter and g is the gap function. The equilibrium

configuration is obtained by solving for the stationary condition of the following

equation with respect to displacement u and contact pressure λN ,

Πlag(u, λN) =2∑i=1

Π(i)(u(i)) +

∫Γ

(i)c

λNgdΓi ∀ wi ∈ V (3.35)

By taking a directional derivative of eq.(3.35),

Du Πlag.w =2∑i=1

G(i)(u(i),w(i)) +

∫Γ

(i)c

λNδg dΓ (3.36)

∫Γ

(i)c

(λN − qN)g dΓ ≥ 0 (3.37)

for all admissible variations qN ∈M of λN , where

M = qN : Γ(i)c → R+

∣∣∣∣ qN ≥ 0 (3.38)

The solution of a lagrange multiplier contact problem is a saddle point, i.e. a point

corresponding to minimum of Πlag with respect to u and a maximum of Πlag with

respect to λN, which is obtained by solving the eq.(3.36) and (3.37). As the function

qN is constrained qN ≥ 0 , it is difficult to implement the above formulation in a

finite element framework. An equivalent statement of the above equation is provided,

which can be implemented numerically.

Gint,ext(u,w) +

∫Γ

(i)c

λN(x) δg(x) dΓ(i) = 0 (3.39)

22

subject to Kuhn-Tucker optimality conditions of

λN ≥ 0

g ≤ 0

λNg = 0 (3.40)

Eq.(3.40) is restatement of eq.(3.18) where the Lagrange parameter λN corresponds

to contact pressure tN .

3.2.2 Penalty Method

The penalty method reduces the contact problem to an unconstrained optimization

problem, by removing the constraints explicitly from the variational formulation with

a contact functional, defined as

Πc =

∫Γ

(i)c

εN〈g〉2 dΓi (3.41)

where, εN > 0 is the penalty parameter and

〈g〉 =

g g ≥ 00 g ≤ 0

(3.42)

The total potential of the two body system can be written as,

Πpen(u) =2∑i=1

Π(i)(u(i)) +

∫Γic

εN〈g〉2 dΓ (3.43)

Applying a directional derivative to the above equation, we get

Du Πpen.w = Gint,ext +

∫Γ

(i)c

εN〈g〉δg dΓ = 0 (3.44)

Eq.(3.44) is then discretized using finite dimensional solution and weighting spaces

and assembled to form matrix equations. This is then solved to obtain the dis-

placement field. A comparison of eq.(3.34) and eq.(3.41) shows that the Lagrange

23

Figure 3.2: Contact traction vs gap [17]

multiplier is equivalent to the penalty force, tN = εN < g >. The pressure gap

relationship tN = εN < g > is as shown in fig.(3.2). The impenetrability condition

can only be achieved if εN →∞, which introduces ill-conditioning in numerical cal-

culations. Penalty method eliminates the need for solving the problem with respect

to two variables at the expense of imposing the contact constraint approximately

rather then exactly.

3.2.3 van der Waals forces

Any two atoms of the same material or two different materials interact with each

other as shown in fig.(3.3). When these two atoms are far apart from each other,

their behavior is influenced by an attractive force and when they are close to each

other, their interaction is dominated by a strong repulsive force. This interaction at

the atomistic level prevents any two bodies from interpenetrating in the real world.

The attractive - repulsive characteristic is modeled using the Lennard-Jones(L-J)

potential [29]. The L-J potential defines the potential variation between the two

atoms as a function of r, where r is the distance between the two atoms. The L-J

potential models the pair interaction, i.e. the interaction between any two atoms. In

24

Figure 3.3: L-J potential vs r/r0 [29]

this work, three and more atoms interaction is not considered. The L-J potential is

given as,

φ(r) = ε(r0

r

)12

− 2ε(r0

r

)6

(3.45)

where r0 is the equilibrium distance between the atoms and ε is the energy of the

well at r = r0. The force between them is given as the negative gradient of the L-J

potential with respect to r,

F = −∂φ∂r

(3.46)

=12ε

r0

[(r0

r

)13

−(r0

r

)7]

(3.47)

The first term in the eq.(3.47) represents the repulsive region and second term

contributes to the attractive region of the curve in fig.(3.3). Usually, the equilibrium

distance r0 is of the order of nm. According to the L-J potential, this attractive force

tends to zero for a r/r0 ratio > 1000. Thus, at micro to macro scale the effect of

the L-J potential is negligible. The contact interaction between Ω1 and Ω2 can be

obtained by integrating the L-J potential over the domain and can be written as,

Πc =

∫Ω1

∫Ω2

β1β2φ(r)dΩ1dΩ2 (3.48)

25

Figure 3.4: L-J potential aura of two bodies

where, β1, β2 are the number densities of molecules in Ω1, Ω2. The variation of the

contact potential energy can be derived as,

δΠc =

∫Ω1

∫Ω2

β1β2

(∂φ(r)

∂x1

δφ1 +∂φ(r)

∂x2

δφ2

)dΩ1dΩ2 (3.49)

= −∫

Ω1

δφ1β1b1dΩ1 −∫

Ω2

δφ2β2b2dΩ2 (3.50)

where, b1,b2 are given as,

b1 =

∫Ω2

β2F (r)rdΩ2 b2 =

∫Ω1

β1F (r)rdΩ1 (3.51)

As shown in fig.3.4, each body has an aura which affects the neighboring bodies and

creates a body force field in them. b1 is the body force field exerted by the body 2

on body 1 and b2 is the body force field exerted by the body 1 on body 2(fig.3.4).

Eq.(3.51) is integrated analytically with the assumption that the mask and chuck

surfaces are flat planes and the deformation of the system is small enough that the

number densities are constant. The closed form result of eq.(3.51) is as below,

b1 = πβ2εr02

[1

5

(r0

r

)10

−(r0

r

)4]

(3.52)

b2 = πβ2εr02

[1

5

(r0

r

)10

−(r0

r

)4]

(3.53)

26

The variation of the contact potential given by eq.(3.49) can be integrated analyt-

ically by substituting eq.(3.52) and eq.(3.53). The volume integration is first done

by integrating along the thickness of the mask and chuck, with upper limit tending

to infinity [31]. The aura of body 1 exerts a significant amount of force on body 2

only up to 1000r0 distance inside the contact surface. As r0 is of the order of nm,

considerable amount of force due to van der Waals effect is experienced only to the

order of micrometer inside the body from the contact surface. Thus the integration

of L-J potential along the thickness with an upper limit of infinity is justified.

δΠc = −∫

Γ01

δφ1

∫ ∞g

β1b1drdA1 −∫

Γ02

δφ2

∫ ∞g

β2b2drdA2 (3.54)

δΠc = −∫

Γ01

δφ1.t1dA1 −∫

Γ02

δφ2.t2dA2 (3.55)

As the contact tractions acting on the contact surfaces are opposite and equal,

eq.(3.52) can be written in a compact form as,

δΠc = −∫

Γ01

(δφ1 − δφ2) .t1dA1 (3.56)

The contact traction acting on Γc due to van der Waals forces are derived to be,

t1 =

∫ ∞g

β1b1dr (3.57)

= β1β2πεr03

[1

45

(r0

g

)9

− 1

3

(r0

g

)3]

(3.58)

The implementation of this contact traction as a function of gap in FEAP is validated

in next chapter.

27

4

Effect of chucking on nonflatness - Numericalresults

In this chapter, numerical results obtained from electrostatic chucking simulation for

three different waviness profiles of the reticle and chuck surface are presented. We

first validate the van der Waals force contact constraint implementation in FEAP

8.1 with a simple two half space interaction problem. A brief explanation of the

loading methodology to simulate the chucking process is given. We then present the

final chucked surface shapes of the mask front and back side at 10000V force. The

variation of peak-valley of the mask front side with voltage is provided for all three

sets of nonflatness data [6].

4.1 Validation of van der Waals force implementation in FEAP

The interaction of two half space problem with applied voltage is considered to

validate the implementation of the van der Waals interaction. In this problem, one

of the half space is considered to be rigid and the other as deformable. As shown in

Fig.4.1(a), the deformable half space is modeled as a spring with equivalent stiffness,

k = EAL

where, E is the Young’s modulus, A is the area of cross section and L is

28

the length of half space. The half spaces are separated by an initial gap of g0. This

problem is driven by an applied voltage which is a function of the gap between the

two half spaces and the voltage itself. The impenetrability constraint is enforced

through van der Waals forces. The force equilibrium of this system is given by,

k1u1 = Fvw + Fev (4.1)

where Fvw is the van der Waals force between the two half planes, which can be

derived analytically from eq.(3.45) and is given as

Fvw = −β1β2πεr30A

[1

45

(r0

g

)9

− 1

3

(r0

g

)3]

(4.2)

where gap, g = g0−u1. In this derivation, the half plane was assumed to be infinitely

long to evaluate the double volume integral. This assumption is justified by the use

of Leonard Jones potential for the van der Waals force interaction. L-J potential as

explained in chapter 3, is strong at nm scale and weakens out at micro and macro

scale. Electrostatic force can be obtained from eq.(2.7),

Few =1

8ε0

[k

(kg + d)

]2

V 2A (4.3)

An equivalent finite element model is built in FEAP as shown in Fig.4.1(b). Body

1 discretized with a single solid element is fixed at all nodes to model the rigid

half space. Body 2 of size 5 × 5 ×12 mm is meshed with 5 solid elements and

assigned a Young’s modulus of 67.6 GPA. Body 2 is constrained at four nodes on

the top surface as shown in Fig.4.1(b). This simulation was run for different voltage

and the gap between the two body 1 and 2 was monitored. Fig.4.2. shows the

variation of the displacement of the free end of spring with applied voltage. A good

comparison between the exact and finite element solution is observed. As a load

incremental scheme was used, the dip in the voltage vs displacement ratio curve was

not captured.

29

(a) Spring half space model (b) Finite Element modelFigure 4.1: Modeling details

Figure 4.2: Comparison of exact vs FE solution

30

4.2 Effect of Chucking

In this section, finite element simulations are run with three different nonflatness

data of the mask and chuck. The effect on the P-V of the chucked mask surface is

analyzed with increase in voltage. In the manufacturing line, the mask is placed on

the chuck using automated controls. As explained in chapter 2, electrostatic chucking

is used to avoid any contacts on the mask frontside. So during the chucking process,

the mask front side is free of any constraint. To simulate the chucking process with

a free mask frontside, the following loading methodology is used. For the penalty

constraint,

1. The mask frontside is fixed at four corner nodes with an initial gap between

the mask backside and chuck frontside.

2. The simulation is run for say, 200V to initiate the contact between the mask

and chuck with a coefficient of friction of 1.0.

3. The boundary condition on the frontside of the mask is then removed. The

contact surface between the mask and chuck is always in the stick state which

avoids the rigid body motion of the mask.

4. The analysis is then continued until 10000V with a load increment strategy.

For the van der Waals constraint,

1. The mask frontside is fixed at four corner nodes with an initial gap between

the mask backside and chuck frontside.

2. van der Waals forces provides the normal contact constraint. For the tangential

contact, Coulomb friction was used with penalty regularization. The simulation

is then run for 200V to ensure atleast one of the contact nodes is in the repulsive

31

region of the van der Waals force curve. The friction is initiated when any of

the contact nodes enters the repulsive region. A high coefficient of friction

(1e6) is used to model the stick condition of the mask - chuck interface

3. The boundary condition on the frontside of the mask is then removed and

reaction forces are noted. These reaction forces are applied to the freed bound-

ary nodes of the mask to ensure stability of the simulation. These reaction

forces are then gradually reduced to 0 with increase in voltage. This gradual

reduction in voltage is achieved before 800V force.

4. The analysis is then continued until 10000V with a load increment strategy.

The three different nonflatness reticle data namely, Reticle 070425, Reticle 070622

and Reticle 080408 are modeled and analyzed. Initial frontside and backside surface

profile of Reticle 070622 is flatter when compared to Reticle 070425 and Reticle

080408. Reticle 080408 has a comparable initial profile with a backside maximum P-

V value of 1.128769 µm and frontside maximum P-V value of 1.166010 µm. Though

Reticle 070425 has a similar initial backside and frontside profile, their max P-V

value differs by 0.9 µm.

4.2.1 Nonflatness data 1 - Reticle 070425

This section provides the results obtained with Reticle 070425 nonflatness data of the

mask and chuck surface. Fig. 4.3(a) and 4.4(a) shows the initial nonflat surface of the

mask backside and frontside respectively. The initial peak-valley(P-V) of the mask

backside and the mask front side is 1.525813 µm and 0.588550 µm respectively. The

final chucked mask backside and frontside are as shown in Fig.4.3(b) and Fig.4.4(b)

at a 10000 voltage force. A corresponding 2D cross-section of these profiles at x = 0

plane are shown in Fig.4.5. This plot shows that the flattening of the mask backside

affects the mask frontside. The P-V of a chucked mask frontside is expected to

32

−60−40

−200

2040

60

−50

0

50

0

5

10

15

x 10−4

x−axis (mm)

Warped Backside Mask −Before Chucking, Peak−Valley = 1.525813µm

y−axis (mm)

heig

ht (

mm

)

2

4

6

8

10

12

14

x 10−4

(a) Before Chucking

−60−40

−200

2040

60

−50

0

50

−2

−1

0

1

2

3

x 10−4

x−axis (mm)

Mask Backside − After Chucking, 10000V, Peak−Valley = 0.189896µm

y−axis (mm)

heig

ht (

mm

)

−8

−6

−4

−2

0

2

4

6

8

10

x 10−5

(b) van der Waals - After Chucking

−60−40

−200

2040

60

−50

0

50

−3

−2

−1

0

1

2

x 10−4

x−axis (mm)

Chucked Mask Backside, 10000V, Peak−Valley = 0.192846µm

y−axis (mm)

heig

ht (

mm

)

−10

−8

−6

−4

−2

0

2

4

6

8

x 10−5

(c) Penalty - After ChuckingFigure 4.3: Mask Backside Nonflatness - Reticle 070425

decrease with increase in voltage. Due to the influence of the initial waviness profile

of the chuck frontside and mask backside, an increase in the P-V of the chucked

mask frontside is observed with increase in voltage. Qualitatively, the mask backside

conforms to the chuck frontside profile, as the chuck is stiffer than the mask. As seen

from the Fig.4.3, the mask backside P-V reduces from 1.525813 µm to 0.192846 µm

with the penalty method. This flattening of the mask backside of the order of 1.3

µm causes the reversal in sign of the curvature of the mask frontside waviness profile

and an increase in the final P-V value as shown in Fig.4.4(b).

The waviness(∼ 100nm) at the mask backside is observed to play an important

33

−60−40

−200

2040

60

−50

0

50

6.5004

6.5006

6.5008

6.501

6.5012

x−axis (mm)

Warped Frontside Mask −Before Chucking, Peak−Valley = 0.588550µm

y−axis (mm)

heig

ht (

mm

)

6.5005

6.5006

6.5006

6.5007

6.5007

6.5008

6.5008

6.5009

6.5009

6.501

6.501

(a) Before Chucking

−60−40

−200

2040

60

−50

0

50

6.4995

6.5

6.5005

x−axis (mm)

Mask Frontside − After Chucking, 10000V, Peak−Valley = 1.023193µm

y−axis (mm)

heig

ht (

mm

)

6.4996

6.4997

6.4998

6.4999

6.5

6.5001

6.5002

6.5003

6.5004

6.5005

6.5006

(b) van der Waals - After Chucking

−60−40

−200

2040

60

−50

0

50

6.4995

6.5

6.5005

x−axis (mm)

Chucked Mask Frontside, 10000V, Peak−Valley = 1.040595µm

y−axis (mm)

heig

ht (

mm

)

6.4996

6.4997

6.4998

6.4999

6.5

6.5001

6.5002

6.5003

6.5004

6.5005

6.5006

(c) Penalty - After ChuckingFigure 4.4: Mask Frontside Nonflatness - Reticle 070425

role in the final waviness of the mask frontside though the mask thickness(6.35mm)

is of the order of mm.

Fig.4.6 shows the variation of the P-V of the mask frontside with voltage for

penalty and van der Waals force method. The finite element model was run for

different penalty parameters to remove the penalty sensitivity. As shown in the

Fig.4.6, the model is penalty sensitive to the order of 0.1 nm. The van der Waals

and penalty method compare well to an order of 4 - 15 nm accuracy for Reticle

070425.

34

−80 −60 −40 −20 0 20 40 60 800.5

1

1.5

2

2.5

3x 10

−3

y−axis (mm)

z−ax

is (

mm

)

Reticle 070425 Before Chucking, x=0 cross−section

Mask FrontsideMask Backside

(a) Before Chucking

−80 −60 −40 −20 0 20 40 60 800

0.5

1

1.5x 10

−3

y−axis (mm)

z−ax

is (

mm

)

Reticle 070425 After Chucking, x=0 crosssection


(b) van der Waals - After ChuckingFigure 4.5: Reticle 070425 Profile at x=0 crosssection


The discrete mask and chuck surface are warped with Reticle 070622 waviness data.

Fig. 4.7(a) and 4.8(a) shows the initial waviness profile of the mask backside and

mask frontside. Fig.4.8(b) and (c) provides the final mask frontside profile at 10000V

force obtained using van der Waals and penalty method respectively. The P-V com-

Figure 4.6: P-V vs Voltage variation for Reticle 070425

35

pares well between these two methods to the order of 10 nm. The increase in the

P-V of the chucked mask frontside in comparison to the initial P-V is attributed to

the chuck-mask contact interaction as explained for reticle 070425. Fig.4.10 shows

the variation of P-V of mask frontside for various voltages. The P-V compares well

with penalty and van der Waals force method to within 10-12nm accuracy for an

voltage range of 0-10000V. A similar trend was observed for the reticle 070425 data.

Fig.4.6 and 4.10 shows a decrease in P-V with increase in voltage for van der

Waals force method while P-V is constant with penalty method. With the van der

Waals force method, contact nodes can be either in the attractive region or in the

repulsive region. Contact nodes can enter the repulsive region at different levels of

the voltage force. These interactions between the contact node of the mask backside

or chuck frontside, causes the decrease in P-V of mask frontside with an increase in

voltage level. In the penalty method, the interference between the contact surfaces

is constant with any amount of increase in load. Thus the P-V is constant for the

penalty method at higher voltages.


The initial waviness data of the chuck frontside used in all three simulations is the

same with a maximum P-V value of 0.187568 µm. For reticle 080408, the P-V of

the chucked mask frontside is observed to be less than the initial P-V value, as

observed from Fig.4.11(a) and (b). But for reticle 070425 and 070622, the P-V of

the final surface was higher than the initial value. This trend is attributed to the

initial waviness profile of the mask backside and mask frontside(Fig.4.9). For reticle

080408, the P-V of the mask frontside and mask backside are of the same order and

their curvatures have similar signs. With an almost flat chuck front surface, this

mask surface profile tends to reduce the P-V with voltage. The variation of the P-V

of the mask frontside is plotted against voltage in Fig.4.14. The plot shows a good

36

−60−40

−200

2040

60

−50

0

50

8

9

10

11

x 10−4

x−axis (mm)


y−axis (mm)

heig

ht (

mm

)

9.35

9.4

9.45

9.5

9.55

9.6

9.65

9.7

9.75

9.8

9.85

x 10−4

(a) Before chucking

−60−40

−200

2040

60

−50

0

50

−2

−1

0

1

2

3

x 10−4

x−axis (mm)


y−axis (mm)

heig

ht (

mm

)

−6

−4

−2

0

2

4

6

8

10

x 10−5

(b) van der Waals - After chucking

−60−40

−200

2040

60

−50

0

50

−2

−1

0

1

2

x 10−4

x−axis (mm)


y−axis (mm)

heig

ht (

mm

)

−8

−6

−4

−2

0

2

4

6

8

x 10−5

(c) Penalty - After chuckingFigure 4.7: Mask Backside - Nonflatness Reticle 070622

comparison between the penalty and van der Waals method. The difference between

the P-V values obtained from these methods are of the order of 5 nm.

37

−60−40

−200

2040

60

−50

0

50

6.5008

6.5009

6.501

6.5011

6.5012

x−axis (mm)


y−axis (mm)

heig

ht (

mm

)

6.5009

6.5009

6.5009

6.501

6.501

6.501

6.501

6.501

(a) Before chucking

−60−40

−200

2040

60

−50

0

50

6.4998

6.4999

6.5

6.5001

6.5002

6.5003

x−axis (mm)


y−axis (mm)

heig

ht (

mm

)

6.4999

6.5

6.5

6.5001

6.5001


−60−40

−200

2040

60

−50

0

50

6.4998

6.5

6.5002

x−axis (mm)


y−axis (mm)

heig

ht (

mm

)

6.4999

6.4999

6.5

6.5

6.5001

(c) Penalty - After chuckingFigure 4.8: Mask Frontside - Nonflatness Reticle 070622

38

−80 −60 −40 −20 0 20 40 60 800.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8x 10

−3

y−axis (mm)

z−ax

is (

mm

)



(a) Before Chucking

−80 −60 −40 −20 0 20 40 60 80−2

0

2

4

6

8

10

12x 10

−4

y−axis (mm)

z−ax

is (

mm

)





39

−60−40

−200

2040

60

−50

0

50

5

10

15

x 10−4

x−axis (mm)


y−axis (mm)

heig

ht (

mm

)

4

6

8

10

12

14x 10

−4

(a) Before Chucking

−60−40

−200

2040

60

−50

0

50

−2

−1

0

1

2

3

x 10−4

x−axis (mm)


y−axis (mm)

heig

ht (

mm

)

−8

−6

−4

−2

0

2

4

6

8

10

x 10−5


−60−40

−200

2040

60

−50

0

50

−2

−1

0

1

2

x 10−4

x−axis (mm)


y−axis (mm)

heig

ht (

mm

)

−8

−6

−4

−2

0

2

4

6

8

x 10−5

(c) Penalty - After chuckingFigure 4.11: Mask Backside - Nonflatness Reticle 080408

40

−60−40

−200

2040

60

−50

0

50

6.5

6.5005

6.501

6.5015

x−axis (mm)


y−axis (mm)

heig

ht (

mm

)

6.5002

6.5004

6.5006

6.5008

6.501

6.5012

(a) Before Chucking

−60−40

−200

2040

60

−50

0

50

6.4996

6.4998

6.5

6.5002

x−axis (mm)


y−axis (mm)

heig

ht (

mm

)

6.4997

6.4997

6.4998

6.4998

6.4999

6.5

6.5

6.5

6.5001

6.5001


−60−40

−200

2040

60

−50

0

50

6.4996

6.4998

6.5

6.5002

x−axis (mm)


y−axis (mm)

heig

ht (

mm

)

6.4997

6.4997

6.4998

6.4998

6.4999

6.5

6.5

6.5

6.5001

6.5001

(c) Penalty - After chuckingFigure 4.12: Mask Frontside - Nonflatness Reticle 080408

41

−80 −60 −40 −20 0 20 40 60 800.5

1

1.5

2

2.5

3x 10

−3

y−axis (mm)

z−ax

is (

mm

)



(a) Before Chucking

−80 −60 −40 −20 0 20 40 60 800

0.2

0.4

0.6

0.8

1

1.2x 10

−3

y−axis (mm)

z−ax

is (

mm

)





42

5

Conclusion and Future Work

The main focus of this work was to analyze the effect on the peak to valley value of

the nonflatness of the mask frontside with increase in voltage for various nonflatness

data. As the nonflatness are of the order of nanometers, van der Waals forces were

considered in the simulation to model the impenetrability condition between the

mask backside and chuck frontside to replicate the physics. Penalty method of im-

posing the contact constraint was also studied for the ease of running the simulations.

van der Waals forces were implemented in FEAP 8.1. The code was validated with a

two half space interaction problem driven by voltage. The variation in the displace-

ment with voltage compared well between the exact solution and the finite element

simulation. Electrostatic chucking simulations for real mask and chuck were run for

three nonflatness data. To model the chucking process with a free mask frontside,

a loading methodology was developed for van der Waals contact constraint. The

P-V variation with voltage compares well within 10-15 nm accuracy for both the

penalty and van der waals force contact constraint. The initial nonflatness of the

chuck frontside and mask backside was found to play a major role in the final P-V of

the mask frontside. The major contribution of this work in the electrostatic chucking

43

simulation of the mask-chuck system is

1. Including the gap dependency in the electrostatic loading

2. Modeling of van der Waals forces to study the mask-chuck interaction

With van der Waals forces being used to enforce the normal penalty constraint and

tangential penalty to model stick/slip condition, a future research work can be done

in identifying an appropriate theory of non-contact friction and the implementation

of the theory with respect to the finite element framework.

44

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effect of van der waals forces on reticle nonflatness in extreme

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