30 August 2005

On the Modeling of Step-and-Flash ImprintLithography using Molecular Statics Models

Maciej Paszynski a 1 , Albert Romkes a, Elizabeth Collister b,Jason Meiring b, Leszek F. Demkowicz a, and C. Grant Willson b

a Institute for Computational Engineering and Sciences

The University of Texas at Austin

Austin, Texas 78712

bDepartment of Chemical Engineering

The University of Texas at Austin

Austin, Texas 78712

Abstract

We introduce linear and nonlinear molecular statics models for the prediction ofthe material response of polymerized networks in cured etch-barrier layers that areformed during the so-called Step-and-Flash Imprint Lithography (SFIL) process.The molecular structure of the polymerized network is obtained via a Monte-Carlosimulation of the chemical reactions that take place during the curing of the etch-barrier layer. The resulting molecular structure consists of a lattice of point massparticles with pairwise, nearest-neighbor, force interactions governed by potentials.We introduce three molecular statics models: (1) A linear model assuming smalldeformations and inter-molecular bonds governed by quadratic force potential func-tions. (2) A non-linear model allowing for large deformations, still with quadraticforce potential functions. (3) A non-linear model allowing for large deformationsand assuming non-linear Lennard-Jones potential functions. We demonstrate theapplication of these molecular models by showing numerical results on the deforma-tion of a small-size (50×50×50 nm) representative feature in common etch-barrierlayers. These molecular statics models simulations are compared to the results ob-tained by finite element approximations of a linear elasticity model in which theshrinkage of the feature is enforced by the thermal expansion coefficient (CTE).

1 On leave from AGH University of Science and Technology, Department of

Computer Methods in Metallurgy, Cracow, Poland, e-mail: maciek@ices.utexas.edu

1 Introduction

Step and flash imprint lithography (SFIL) is a patterning process utilizingphotopolymerization to replicate the topography of a template onto a sub-strate [2,3].

The major processing steps of SFIL include: depositing a low viscosity, silicon-containing, photocurable etch barrier on to a substrate; bringing the templateinto contact with the etch barrier; curing the etch barrier solution throughUV exposure; releasing the template, while leaving high-resolution featuresbehind; a short, halogen break-through etch; and finally an anisotropic oxygenreactive ion etch to yield high aspect ratio, high resolution features.

Photopolymerization, however, is often accompanied by densification. The in-teraction potential between photopolymer precursors undergoing free radicalpolymerization changes from van der Waals’ to covalent. The average distancebetween molecules decreases and causes volumetric contraction. Densificationof the SFIL photopolymer (the etch barrier) may affect both the cross sectionalshape of the feature and the placement of relief patterns.

The Monte-Carlo algorithm for the simulation of the chemical reactions thattake place during the curing of the etch-barrier layer [6] together with themolecular statics models make it possible to explore the influence of densi-fication and mechanical properties on changes in the placement and in thegeometry of the replicated features.

We introduce linear and non-linear molecular statics models for the predic-tion of the material response of polymerized networks in cured etch-barrierlayers that are formed during the so-called Step-and-Flash Imprint Lithogra-phy (SFIL) process. To obtain physically reasonable realizations of such net-works, a Monte-Carlo algorithm for the simulation of the chemical reactionsthat take place during the curing of the etch-barrier layer was developed [6].The resulting molecular structure consists of a lattice of point mass particleswith pairwise, nearest-neighbor, force interactions governed by force poten-tial functions. In the linear molecular statics model, particle displacementsare assumed to remain small and all inter-molecular bonds are governed byquadratic force potential functions. We additionally introduce nonlinear mod-els that allow for large particle displacements and may employ Lennard-Jonestype potential functions to prescribe the non-covalent force interactions. Wedemonstrate the application of these molecular models by showing numeri-cal results on the deformation of a small-size (50×50×50 nm) representative

ii

feature in common etch-barrier layers.

We compare the molecular statics models with linear elasticity model approx-imated by the finite element method, where the shrinkage of the feature isenforced by the thermal expansion coefficient (CTE).

1

2 The SFIL Process

Fig. 2.1: SFIL process.

The SFIL process can be described in the following steps, according to [3].

• Dispense. The SFIL process employs a template / substrate alignmentscheme to bring a rigid template and substrate into parallelism, trappingthe etch barrier in the relief structure of the template.

• Imprint. The gap is closed until the force that ensures a thin base layer isreached.

• Exposure. The template is then illuminated through the backside to cureetch barrier.

• Separate. The template is withdrawn, leaving low-aspect ratio, high resolu-tion features in the etch barrier.

• Breakthrough Etch. The residual etch barrier (base layer) is etched awaywith a short halogen plasma etch.

• Transfer Etch The pattern is transferred into the transfer layer with ananisotropic oxygen reactive ion etch, creating high-aspect ratio, high reso-lution features in the organic transfer layer.

2

3 Modeling of the Cured Etch Barrier

A Monte Carlo method is used to simulate the chemical reaction processesthat occur during the photo-curing of the etch barrier. The algorithm pro-duces the molecular structure of the polymerized etch barrier in the form of alattice structure of molecules with connecting springs representing the inter-molecular covalent or van der Waals bonds. A comprehensive description ofthis algorithm is given in [6]. To predict the deformation of the obtained latticestructure, we derive linear and nonlinear molecular statics models, describedin the following sections.

3.1 The Molecular Statics Models

In the following, we derive a general equation governing the equilibrium con-figurations of the molecular lattice structures that are produced by the MonteCarlo simulation of the photo-curing process.

In Figure 3.1, we consider an arbitrary pair of bonded molecules in the latticewith indices α and β which have the lattice position vectors pα = (xα, yα, zα)and pβ. The unknown equilibrium position vectors of particles α and β, underthe action of all their intermolecular bonds, are denoted xα = (xα, yα, zα)and xβ, respectively (see Figure 3.1). In addition, the displacements from theinitial position in the lattice to the equilibrium position are represented bythe vectors uα = xα − pα and uβ = xβ − pβ. Let ‖·‖ denote the Euclideannorms in R

3 and let rαβ = ‖xβ − xα‖ be the distance between particles α

and β in initial configuration. Then, the force Fαβ, along the vector xβ − xα,that is acting on the particle α due to the bond between particles α and β isgoverned by the potential function V (rαβ),

Fαβ = −∂V (rαβ)

∂rαβ︸ ︷︷ ︸

magnitude

xβ − xα

‖xβ − xα‖︸ ︷︷ ︸

direction

(1)

If the indices of bonded neighboring particles of particle α are collected in theset Nα, then we obtain its force equilibrium by applying the following sum:

∑

β∈Nα

Fαβ = −∑

β∈Nα

∂V (rαβ)

∂rαβ

xβ − xα

‖xβ − xα‖= 0. (2)

3

Fig. 3.1: Bonded pair of molecules.

The characteristics of the potential functions {V (rαβ)}β∈Nαare provided by

the Monte Carlo simulation algorithm discussed in [6] and determine the char-acter of the bonds, i.e. whether they are covalent or van der Waals bonds.Based on different assumptions on the force potential functions and the de-gree of deformation, we derive one linear and two nonlinear Molecular Staticsmodels from (2) that govern the equilibrium positions {xα} of the particles inthe deformed polymerized network.

3.1.1 Linear Model

The linear Molecular Statics model is based on two basic assumptions:

(1) All the force potential functions V (rαβ) are quadratic and therefore themagnitude of the intermolecular forces Fαβ is linearly dependent on rαβ,i.e. (1) becomes

Fαβ = kαβ

(

rαβ + ∆rαβ − r0αβ

)

where rαβ + ∆rαβ is length of the spring in equillibrium configuration ofthe lattice structure and r0

αβ is the unstretched length of the spring (whenFαβ = 0).

(2) Small deformations are assumed. So that the direction of the spring forcesis aligned with the initial vectors pβ − pα.

In the initial configuration, the spring is aligned along pα−pβ. In the equilib-rium configuration it is aligned along xβ − xα. The angle between these twoalignments is governed by

cos θ =(pβ − pα) · (xβ − xα)

‖pβ − pα‖‖xβ − xα‖=

(pβ − pα) · (xβ − xα)

rαβ · (rαβ + ∆rαβ)

4

Since we assumed small deformations

1 ≈(pβ − pα) · (xβ − xα)

rαβ · (rαβ + ∆rαβ)

we get

rαβ + ∆rαβ ≈1

rαβ

(pβ − pα) · (xβ − xα)

The magnitude of the force in the spring between particles α and β becomesnow

Fαβ = kαβ(rαβ + ∆rαβ − r0αβ) =

kαβ

rαβ

(pβ − pα) · (xβ − xα) − kαβr0αβ

The force vector Fαβ is approximately aligned along pβ − pα, i.e.

Fαβ = [kαβ

rαβ

(pβ − pα) · (xβ − xα) − kαβr0αβ]

(pβ − pα)

rαβ

where [kαβ

rαβ(pβ−pα) ·(xβ−xα)−kαβr0

αβ] represents the magnitude and(pβ−pα)

rαβ

the direction of the force.

Equilibrium equations for linear model

At each particle, force equilibrium has to hold. Thus

∑

β

Fαβ = 0, α = 1, 2, ..., N

which can be rewritten as

∑

β

(kαβ

rαβ

(pβ − pα) · (xβ − xα))pβ − pα

rαβ

=∑

β

kαβr0αβ

rαβ

(pβ − pα) = 0

Note that rαβ =√

(xβ − xα)2 + (yβ − yα)2 + (zβ − zα)2. So at each particle we

have 3 equations (corresponding to each component of the vector equation).

For example, equations in x direction

∑

β

kαβ[(xβ − xα)(xβ − xα) + (yβ − yα)(yβ − yα) + (zβ − zα)(zβ − zα)](xβ − xα)

(xβ − xα)2 + (yβ − yα)2 + (zβ − zα)2=

∑

β

kαβr0αβ(xβ − xα)

√

(xβ − xα)2 + (yβ − yα)2 + (zβ − zα)2

equations in y direction

∑

β

kαβ[(xβ − xα)(xβ − xα) + (yβ − yα)(yβ − yα) + (zβ − zα)(zβ − zα)](yβ − yα)

(xβ − xα)2 + (yβ − yα)2 + (zβ − zα)2=

5

∑

β

kαβr0αβ(yβ − yα)

√

(xβ − xα)2 + (yβ − yα)2 + (zβ − zα)2

and in z direction

∑

β

kαβ[(xβ − xα)(xβ − xα) + (yβ − yα)(yβ − yα) + (zβ − zα)(zβ − zα)](zβ − zα)

(xβ − xα)2 + (yβ − yα)2 + (zβ − zα)2=

∑

β

kαβr0αβ(zβ − zα)

√

(xβ − xα)2 + (yβ − yα)2 + (zβ − zα)2= 0

3.1.2 Nonlinear models

There are two non-linear models under consideration.

(1) In the first non-linear model, the intermolecular forces Fαβ are still linearbut large deformations are assumed. It implies that directions of the springforces are now aligned with xβ −xα. Thus, the non-linearity of the model is aconsequence of large deformations.

(2) In the second non-linear model we allow for large deformations and all theforce potential functions V (rαβ) are assumed to be non-linear, of the Lennard-Jones type [4]

V (rαβ) = Cαβ[

(

σαβ

rαβ

)nαβ

−

(

σαβ

rαβ

)mαβ

] (3)

Thus, for each spring we need data on the four parameters Cαβ, σαβ, mαβ andnαβ.

Equilibrium equations for the non-linear model with quadratic potentials

At each particle α, force equilibrium has to hold. Thus

∑

β

Fααβ = 0, α = 1, 2, ..., N

which can be rewritten as

F(X) = 0

where F is a composition of Fα for all α = 1, 2, ..., N , and X is a compositionof x1, x2, ..., xN .

Using a Taylor expansion of F around a guess vector Xi we have

F(X) = F(Xi) + ∂F(Xi, ∆Xi) + H.O.T. (4)

6

where X = Xi + ∆Xi and H.O.T. is an abreviation for higher order terms.Since

F(X) ≈ F(Xi) + ∂F(Xi, ∆Xi). (5)

then ∆Xi can be approximated from

∂F(Xi, ∆Xi) = −F(Xi). (6)

and

F(Xi) =∑

β

Fαβ(xiα,xi

β) =∑

β

kαβ(‖xiβ − xi

α‖ − r0αβ) ·

xiβ − xi

α

‖xiβ − xi

α‖

Here xiα = (xi

α, yiα, zi

α) and ‖xiα‖ =

√

(xiα)2 + (yi

α)2 + (ziα)2.

We now derive the formula for the Jacobian

∂F(Xi, ∆Xi) =∑

β

∂Fαβ(xiα,xi

β; ∆xiα, ∆xi

β)

By definition ∂Fαβ(xiα,xi

β; ∆xiα, ∆xi

β) is given by

limθ→0

1

θ[Fαβ(xi

α + θ∆xiα,xi

β + θ∆xiβ) − Fαβ(xi

α,xiβ)]

= limθ→0

1

θ[kαβ(‖xi

β + θ∆xiβ − xi

α − θ∆xiα‖ − r0

αβ)xi

β + θ∆xiβ − xi

α − θ∆xiα

‖xiβ + θ∆xi

β − xiα − θ∆xi

α‖

−kαβ(‖xiβ − xi

α‖ − r0αβ)

xiβ − xi

α

‖xiβ − xi

α‖]

= limθ→0

[kαβ(‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖ − r0αβ)]

∆xiβ − ∆xi

α

‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖

+ limθ→0

1

θ[kαβ‖x

iβ − xi

α + θ(∆xiβ − ∆xi

α)‖]xi

β − xiα

‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖

− limθ→0

1

θkαβ[‖xi

β − xiα‖]

xiβ − xi

α

‖xiβ − xi

α‖

− limθ→0

1

θkαβr0

αβ[1

‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖−

1

‖xiβ − xi

α‖](xi

β − xiα)

= kαβ[‖xiβ − xi

α‖ − r0αβ]

∆xiβ − ∆xi

α

‖xiβ − xi

α‖

− limθ→0

1

θkαβr0

αβ[‖xi

β − xiα‖

2 − ‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖2

‖xiβ − xi

α‖ + ‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖]

·xi

β − xiα

‖xiβ − xi

α‖ · ‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖

(7)

7

= kαβ[‖xiβ − xi

α‖ − r0αβ]

∆xiβ − ∆xi

α

‖xiβ − xi

α‖

− limθ→0

1

θkαβr0

αβ[−2θ(xi

β − xiα) · (∆xi

β − ∆xiα) − θ2‖∆xi

β − ∆xiα‖

2

‖xiβ − xi

α‖ + ‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖]

·xi

β − xiα

‖xiβ − xi

α‖ · ‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖

= kαβ[‖xiβ − xi

α‖ − r0αβ]

∆xiβ − ∆xi

α

‖xiβ − xi

α‖+ kαβr0

αβ[(xi

β − xiα) · (∆xi

β − ∆xiα)

‖xiβ − xi

α‖2

] ·xi

β − xiα

‖xiβ − xi

α‖

(8)

We can reformulate the equations in matrix-vector notation, i.e.

∂Fαβ(Xi; ∆Xi) =(

Kαα Kαβ)

∆xiα

∆xiβ

(9)

In the following, we use Einstein’s notation aibi = a1b1 +a2b2 +a3b3. We definethe Cartesian base vectors e1 = (1, 0, 0), e2 = (0, 1, 0), and e3 = (0, 0, 1).Hence b = biei and a · b = (aiei) · (bjej) = aibj(ei · ej) = aibjδij = aii. Nowwe can write

∆xiα = ∆ξi

α,jej xiα = ξi

α,jej

∆xiβ = ∆ξi

β,jej︸ ︷︷ ︸

xiβ = ξi

β,jej︸ ︷︷ ︸

unknown known

Furthermore, for simplicity in the notation, we define

Φiαβ = kαβ

[‖xiβ − xi

α‖ − r0αβ]

‖xiβ − xi

α‖

and

ϕiαβ =

kαβr0αβ

‖xiβ − xi

α‖3

8

Now, we can rewrite equation (8)

∂Fαβ = Φiαβ(∆ξi

β,j − ∆ξiα,j)ej + ϕi

αβ[(ξiβ,k − ξi

α,k)ek · (ξiβ,l − ξi

α,l)el](ξiβ,j − ξi

α,j)ej

=(

Φiαβ(∆ξi

β,j − ∆ξiα,j) + ϕi

αβ[ξiβ,l∆ξi

β,l − ξiβ,l∆ξi

α,l − ξiα,l∆ξi

β,l + ∆ξiα,lξ

iα,l] × (ξi

β,j − ξiα,j)

)

ej

=

−[Φiαβ + ϕi

αβ(ξiβ,j − ∆ξi

α,j)2]∆ξi

α,j −∑

l 6=j

ϕiαβ(ξi

β,l − ξiα,l)(ξ

iβ,j − ξi

α,j)∆ξiα,l

+ [Φiαβ + ϕi

αβ(ξiβ,j − ξi

α,j)2]∆ξi

β,j +∑

l 6=j

ϕiαβ(ξi

β,l − ξiα,l)(ξ

iβ,j − ξi

α,j)∆ξiβ,l

ej

(10)

Thus

Kαα =

−(Φiαβ + ϕi

αβ(xiβ − xi

α)2) −ϕiαβ(yi

β − yiα)(xi

β − xiα) −ϕi

αβ(ziβ − zi

α)(xiβ − xi

α)

−ϕiαβ(xi

β − xiα)(yi

β − yiα) −(Φi

αβ + ϕiαβ(yi

β − yiα)2) −ϕi

αβ(yiβ − yi

α)(ziβ − zi

α)

−ϕiαβ(xi

β − xiα)(zi

β − ziα) −ϕi

αβ(yiβ − yi

α)(ziβ − zi

α) −(Φiαβ + ϕi

αβ(ziβ − zi

α)2)

where submatrices are located in α and β rows and columns.

K =

Kαα Kαβ

Kβα Kββ

and Kαβ = −Kαα, Kββ = Kαα, Kβα = Kαβ.

Equilibrium equations for the non-linear model with Lennard-Jones potentials

At each particle α, force equilibrium has to hold. Thus

Rα =∑

β

Fααβ = 0, α = 1, 2, ..., N

9

Collection of all resultant forces is

F =

R1

R2

...

RN

= 0

where F = F(X) and X is a vector in RN with components x1, x2, ..., xN .

The force exerted by all neighboring particles β on a particle α is given by

F(Xi) =∑

β

Fαβ(xiα,xi

β) =∑

β

|Fαβ|xi

β − xiα

‖xiβ − xi

α‖

where

|Fαβ| = −∂V

∂r

and V (r) is the Lennard-Jones potential function given by (3).

Since∂V (r)

∂r= Cαβ[

mαβ

σαβ

(σαβ

r

)mαβ−1

−nαβ

σαβ

(σαβ

r

)nαβ−1

]

thus

Rα(Xi) = −∑

β

Cαβ

σαβ

[mαβ

(

σαβ

‖xiβ − xi

α‖

mαβ+1)

−nαβ

(

σαβ

‖xiβ − xi

α‖

nαβ+1)

]×xi

β − xiα

‖xiβ − xi

α‖

We will compute the Jacobian ∂F. Since

∂F =

∂R1

...

∂RN

and

∂Rα =∑

β

∂Fαβ

then we need to compute

∂Fαβ(∂Xi, ∆∂Xi) = limθ→0

1

θ

(

Fαβ(∂Xi + θ∆∂Xi) − Fαβ(∂Xi))

We will focus on Fαβ(∂Xi + θ∆∂Xi) first.

10

limθ→0

1

θFαβ(∂Xi + θ∆∂Xi)

= − limθ→0

1

θ

(

Cαβ

σαβ

[mαβ(σαβ

‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖)mαβ+1

−nαβ(σαβ

‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖)nαβ+1]

xiβ − xi

α + θ(∆xiβ − ∆xi

α)

‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖

)

= − limθ→0

1

θ

(

Cαβ

σαβ

[mαβ(σαβ

‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖)mαβ+1

−nαβ(σαβ

‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖)nαβ+1]

xiβ − xi

α

‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖

− Cαβσαβ[mαβ(σαβ

‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖)mαβ+1

−nαβ(σαβ

‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖)nαβ+1]

∆xiβ − ∆xi

α

‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖

)

(11)

= limθ→0

Cαβmαβσmαβ

αβ

θ

(

‖xiβ − xi

α‖

‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖mαβ+1

−‖xi

β − xiα + θ(∆xi

β − ∆xiα)‖

‖xiβ − xi

α‖mαβ+1

)

xiβ − xi

α

‖xiβ − xi

α‖‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖

(12)

+ limθ→0

Cαβnαβσnαβ

αβ

θ

(

‖xiβ − xi

α‖

‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖nαβ+1

−‖xi

β − xiα + θ(∆xi

β − ∆xiα)‖

‖xiβ − xi

α‖nαβ+1

)

xiβ − xi

α

‖xiβ − xi

α‖‖xiβ − xi

α + θ(∆xiβ − ∆xi

α)‖

(13)

Let us focus on term (12) now

limθ→0

Cαβmαβσmαβ

αβ

θ

(

‖xiβ − xi

α‖mαβ+2 − ‖xi

β − xiα + θ(∆xi

β − ∆xiα)‖mαβ+2

)

×xi

β − xiα

‖xiβ − xi

α‖mαβ+2‖xi

β − xiα + θ(∆xi

β − ∆xiα)‖mαβ+2

(14)

11

= limθ→0

Cαβmαβσmαβ

αβ

θ

(

‖xiβ − xi

α‖2(mαβ+2) − ‖xi

β − xiα + θ(∆xi

β − ∆xiα)‖2(mαβ+2)

)

×xi

β − xiα

‖xiβ − xi

α‖mαβ+2‖xi

β − xiα + θ(∆xi

β − ∆xiα)‖mαβ+2

×1

[‖xiβ − xi

α‖mαβ+2 + ‖xi

β − xiα + θ(∆xi

β − ∆xiα)‖mαβ+2]

(15)

= limθ→0

Cαβmαβσmαβ

αβ

θ

(

‖xiβ − xi

α‖2(mαβ+2)

− (‖xiβ − xi

α‖2 + 2θ(∆xi

β − ∆xiα) · (xi

β − xiα) + θ2‖∆xi

β − ∆xiα‖

2)mαβ+2

)

×xi

β − xiα

‖xiβ − xi

α‖mαβ+2‖xi

β − xiα + θ(∆xi

β − ∆xiα)‖mαβ+2

×1

[‖xiβ − xi

α‖mαβ+2 + ‖xi

β − xiα + θ(∆xi

β − ∆xiα)‖mαβ+2]

(16)

We can cancel out terms, and take the limit

= limθ→0

Cαβmαβσmαβ

αβ

θ

(

‖xiβ − xi

α‖2(mαβ+2) − ‖xi

β − xiα‖

2(mαβ+2)

− 2θ(mαβ + 2)‖xiβ − xi

α‖2mαβ+3(xi

β − xiα) · (∆xi

β − ∆xiα) + ©(θ2)

)

×xi

β − xiα

‖xiβ − xi

α‖mαβ+2‖xi

β − xiα + θ(∆xi

β − ∆xiα)‖mαβ+2

×1

[‖xiβ − xi

α‖mαβ+2 + ‖xi

β − xiα + θ(∆xi

β − ∆xiα)‖mαβ+2]

(17)

= Cαβmαβσmαβ

αβ

(

2(mαβ + 2)‖xiβ − xi

α‖2mαβ+3(xi

β − xiα) · (∆xi

β − ∆xiα))

×xi

β − xiα

2‖xiβ − xi

α‖3(mαβ+2)

(18)

= Cαβmαβ(mαβ + 2)σmαβ

αβ

(xiβ − xi

α) · (∆xiβ − ∆xi

α)

‖xiβ − xi

α‖mαβ+3

(xiβ − xi

α)

=Cαβmαβ(mαβ + 2)

σ3αβ

(

σαβ

‖xiβ − xi

α‖

)mαβ+3

(xiβ − xi

α) · (∆xiβ − ∆xi

α)(xiβ − xi

α)

(19)

Similarly, term (13)

=Cαβnαβ(nαβ + 2)

σ3αβ

(

σαβ

‖xiβ − xi

α‖

)nαβ+3

(xiβ − xi

α) · (∆xiβ − ∆xi

α)(xiβ − xi

α)

(20)

12

Let us focus on Fαβ(∂Xi) now

limθ→0

Fαβ(∂Xi)

= limθ→0

(

Cαβ

σαβ

[mαβ(σαβ

‖xiβ − xi

α‖

mαβ+1

− nαβ(σαβ

‖xiβ − xi

α‖

nαβ+1

)]xi

β − xiα

‖xiβ − xi

α‖

)

(21)

= −Cαβ

σαβ

[mαβ(σαβ

‖xiβ − xi

α‖)mαβ+1 − nαβ(

σαβ

‖xiβ − xi

α‖)nαβ+1]

∆xiβ − ∆xi

α

‖xiβ − xi

α‖(22)

We finally get

∂Fαβ(∂Xi, ∆∂Xi) = limθ→0

1

θ

(

Fαβ(∂Xi + θ∆∂Xi) − Fαβ(∂Xi))

=Cαβ

σ3αβ

(

nαβ(nαβ + 2)(σαβ

‖xiβ − xi

α‖)nαβ+3 − mαβ(mαβ + 2)(

σαβ

‖xiβ − xi

α‖)mαβ+3

)

×(xiβ − xi

α) · (∆xiβ − ∆xi

α)(xiβ − xi

α)

+Cαβ

σ2αβ

[nαβ(σαβ

‖xiβ − xi

α‖)nαβ+2 − mαβ(

σαβ

‖xiβ − xi

α‖)mαβ+2] × (∆xi

β − ∆xiα)

(23)

Equation (23) can be written in matrix-vector form (9)

As in equation (8), we introduce the coefficients

Φiαβ =

Cαβ

σαβ

[nαβ

σαβ

‖xi

β − xiα‖

nαβ+2

− mαβ

σαβ

‖xi

β − xiα‖

mαβ+2

]

and

ϕiαβ =

Cαβ

σαβ

[nαβ(nαβ+2)

σαβ

‖xi

β − xiα‖

nαβ+3

−mαβ(mαβ+2)

σαβ

‖xi

β − xiα‖

mαβ+3

]

Using this notation, equations (23) become

∂Fαβ(Xi; ∆Xi) = Φαβ(∆xiβ−∆xi

α)+ϕ(

(xiβ − xi

α) · (∆xiβ − ∆xi

α))

×(xiβ−xi

α)

which is exactly the same equation we obtained using quadratic potentialfunction.

13

3.2 Numerical analysis of the Molecular Statics Models

We consider a block of the feature containing IN ×JN ×KN cells. The particlesare located at the grid nodes. Thus, the total number of particles is equal to(IN +1)×(JN +1)×(KN +1). We introduce integer (i, j, k) and real (xi, yj, zk)coordinates for each particle pi,j,k = (xi, yj, zk) (see Figure 3.2) Particles are

j

i

k

Fig. 3.2: Integer coordinates (i, j, k) of particle.

connected by bonds, which are modeled either by linear spring or non-linearLennard-Jones force potential functions. The equilibrium equations are com-puted at each particle by considering the sum of the forces that are exertedby each neighboring particle, see Figure 3.2. Homogeneous displacements areapplied for particles located in the base-plane of the feature.

x xx

x

x

x

x

i,j,k i+1,j,ki−1,j,k

i,j−1,k

i,j+1,k

i,j,k+1

i,j,k−1

x i+1,j+1,kx i−1,j+1,k

xi+1,j−1,kx i−1,j−1,k

xxi−1,j,k+1 i+1,j,k+1

x i+1,j,k−1x i−1,j,k−1

x

x

i,j+1,k+1

i,j−1,k−1

x i,j+1,k−1

x i,j+1,k+1

Fig. 3.3: Each particle interacts with its 27 neighbors.

14

Fig. 3.4: Initial configuration of the grid.

Here, the numerical results are presented for three Molecular Statics Modelsin the case of two configurations of the feature: before and after the template

is released.

Linear model with quadratic potentials, assuming small displacements

We have performed numerical simulation of the deformation of the featureinside the template. The grid of 50 × 50 × 50 particles has been embeddedinside additional layer of particles representing the template. Thus the totalnumber of particles is 52×52×52. The boundary conditions for the model areno displacement of particles located at base of the feature, and no displacementof template particles. The forces between the template particles and the featureparticles describe interactions between the template and the feature.

The initial configuration of the grid is presented in Figure 3.4. The distributionof inter-particle potentials within the grid are obtained from a Monte-Carlosimulation of polymerization process [6].

We will consider here that the small displacement assumption is violated assoon as the displacements exceed ten percent of the distance between neigh-boring particles, in which case the results are deemed unphysical.

We have used the Preconditioned Conjugated Gradients (PCG) from theSLATEC library [7], with diagonal preconditioner. The total computationstime is 87 second. The results are shown in Figure 3.5. We observe relativelysmall displacements of particles.

The next simulation is performed after removal of the template. We haveremoved additional layer of particles representing the template/feature in-teractions. Boundary condition are set to zero displacement at the base. Noexternal forces or boundary conditions are prescribed on the other boundaries

15

Fig. 3.5: Deformation of the feature inside the template provided by the linear model

with quadratic potentials.

Fig. 3.6: Deformation of the feature outside the template provided by the linear

model with quadratic potentials.

of the feature.

The total computational time is 207 second now. Large displacements for thiscase are observed and there is visible shrinkage of the feature. Moreover thefeature has also twisted across the vertical axis of symmetry. This behaviouris unphysical, and we conclude that the linear model is anappropriate for suchsimulations.

Non-linear model with quadratic potentials, allowing for large displacements

In these experiments, the boundary conditions are the same as in the linearmodel discussed above. The difference now is that the non-linearity of themodel requires the Newton-Raphson loop over the iterative solver. We used

16

Fig. 3.7: Convergence history of GMRES solver solving non-linear case with

quadratic potentials, inside template.

Fig. 3.8: Convergence history of GMRES solver solving non-linear case with

quadratic potentials without template.

here GMRES solver rather than PCG solver, as our experiments indicate fasterconvergence .

The convergence history for the simulation with the template is shown inFigure 3.7. The vertical axis denotes absolute error with respect to Eucledeannorm of the displacement vector. Total computational time is 167 seconds.The results of the simulation seem to be similar to those obtained with thelinear model, which can be explained by small displacements occuring in thatcase.

However, the results of non-linear simulation for the case without template areimproved. There is a similar shrinkage of the feature as in the linear modelresults, but there is no twisting of the feature. We conclude that the twist isassociated with the linear model.

The convergence history for the simulation without template is shown in Fig-ure 3.8. Total time is 218 seconds.

17

Fig. 3.9: Deformation of the feature inside the template provided by the non-linear

model with quadratic potentials.

Fig. 3.10: Deformation of the feature outside the template provided by the non-linear

model with quadratic potentials.

Non-linear mixed model with both quadratic and Lennard-Jones potentials, al-

lowing for large displacements

Finally, we have implemented the non-linear model, allowing for mixing quadraticspring potentials for horizontal and vertical bounds, with Lennard-Jones po-tentials for cross-diagonal bounds. We have started the simulation outside thetemplate from initial configuration, presented in Figure 3.4, like for all previ-ously discussed cases. However, we have discovered problems with convergenceof both GMRES and PCG solvers with Newton-Raphson loop.

We have performed the following experiment:

(1) Take the initial grid with parameters of quadratic potentials.(2) Perform computations of non-linear model with quadratic potentials only,

18

Fig. 3.11: Energy space for the non-linear model with quadratic potentials.

allowing for large displacements.(3) Keep resulting configuration of the grid, but replace parameters of po-

tentials into mixed quadratic and Lennard-Jones set.(4) Run computations of non-linear mixed model with both quadratic and

Lennard-Jones potentials

In other words, we have started mixed model computations from results of theprevious model. The GMRES solver stops immediately displaying very smallerror, see Figure 3.13.

From that experiment the following conclusions imply:

• The energy space of the non-linear model with quadratic potentials onlycontains local energy of minimum close to the initial configuration of thefeature, as presented in Figure 3.11. Iterative solvers find only local mini-mum of energy in energy space of the model. We conclude that in the energyspace of the non-linear mixed model there are some local minimum of energyclose to initial configuration, but those local minimums are only some arte-facts of the model. The proper minimum in the mixed model energy spaceis located at the same place as minimum of energy in the non-linear springmodel available from initial configuration. Thus, when we start computa-tions from results of the non-linear spring model, we immediately obtainminimum of energy in the mixed model sense, as illustrated in Figure 3.12.

• There is no need for use of the mixed model, since mixed model results aresimilar to the non-linear spring model results.

19

Fig. 3.12: Energy space for the mixed model with both quadratic and Lennard-Jones

potentials..

Fig. 3.13: Convergence history of GMRES solver solving non-linear case with mixed

quadratic and Lennard-Jones potentials, with initial configuration provided by results

of non-linear model with quadratic potentials.

20

3.3 Numerical analysis of the feature using a continuous linear elas-ticity model

In this section, we model the feature released from the template using linearelasticity.

The bulk properties of the feature are described by the following constitutiverelations

σij = 2µεij + λδijεkk − αδij(3λ + 2µ)(T − T0)

Here α stands for the thermal expansion coefficent (CTE), T and T0 are thecurrent and reference temperatures and ∆T = (T−T0) represents temperaturegradient, µ and λ are the Lame coefficients which, in terms of the Youngmodulus E and Poisson ratio ν are given by

µ =E

2(1 + ν)λ =

νE

1 + ν

where

3λ + 2µ =E

1 − 2νThe model is then specified by providing values of four parameters E, ν,∆T and α. Following measurements presented in [1], [2], [3] we define Youngmodulus for the feature equal to E = 1 GPa, Poisson ratio ν = 0.3, andthermal expansion coefficient as a 10 % volumetric shrinkage of the edge barierfor the temperature gradient ∆T = 1oK as α = ∆V

V ∆T= −0.1.

We have performed finite element of the feature without template, assumingthat the volumetric shrinkage is enforced by the thermal expansion coefficientinvolved in the constitutive equations within linear elasticity model.

The cube-shaped feature with all dimensions equal to 51 × 5.34 nm, fixedat the base, has been deformed in similar way as in the molecular staticscomputations, which is presented in figures 3.14, 3.15 and 3.16.

Figures 3.14 and 3.15 presents symmetric displacements in x and y axis di-rection. The shrinkage of the feature is equal to 14 units, which is less thanthree times the distance between two consecutive particles in the intial con-figuration.

The z-displacement is equal to about 4 distances between particles in theinitial grid configuration, as it is presented in Figure 3.16.

21

Fig. 3.14: Displacements of the feature in x axis direction.

Fig. 3.15: Displacements of the feature in y axis direction.

Fig. 3.16: Displacements of the feature in z axis direction.

22

4 Concluding Remarks

We have introduced three molecular statics models,

• Linear model assuming small deformations and spring potential functions.• Non-linear model allowing for large deformation with spring potential func-

tions.• Non-linear model allowing for large deformations with Lennard-Jones po-

tential functions.

We performed molecular statics simulations for linear model, non-linear quadraticpotential model and mixed quadratic and Lennard-Jones potential model, forboth the feature inside and outside the template.

From these experiments our concluding remarks are

• The results of the linear molecular statics model without the template showshrinkage of the feature with unexpected rotation of the whole feature. Thisis the consequence of the assumption of small displacement, which is notfullfilled in reality.

• The non-linear model with quadratic potentials results in the shrinkage ofthe feature without unexpected rotation. Most of the shrinkage occurs afterthe template is removed.

• The mixed model provides results similar to that of the nonlinear springmodel.

We compared the molecular statics simulation of the feature outside the tem-plate with Finite Element Method computations where the shrinkage of thefeature is enforced by the thermal expansion coefficient (CTE). The FEM re-sults show smaler deformation than the molecular statics results. This may bethe result of the coarse mesh used for the FEM simulation or improper esti-mation of the FEM model parameters. There is no experimental validation ofthe Poisson ratio used.

AcknowledgmentThe first author was supported by the ICES Postdoctoral Fellowship Program.The support of this work, under Semiconductor Research Corporation GrantNo. 2003-MJ-1082, is gratefully acknowledged.

23

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[2] M. Colburn, I. Suez, B. J. Choi, M. Meissl, T. Bailey, S.V. Sreenivasan, J.G. Ekerdt, C. G. Willson, “Characterization and modeling of volumetric andmechanical properties for step and flash imprint lithography photopolymers” J.

Vac. Sci. Technol. B., 19(6), Nov/Dec 2001, pp.2685-2689.

[3] T. Bailey, B. Smith, B.J. Choi, M. Colburn, M. Meissl, S.V. Sreenivasan, J. G.Ekerdt, C. G. Willson, “Step and flash imprint lithography: Defect analysis” J.

Vac. Sci. Technol. B., 19(6), Nov/Dec 2001, pp.2806-2810.

[4] N.M. Putintsev, D.N. Putintsev “Method for Determining the Parameters of theLennard-Jones Potential” Deklady Physical Chemistry, Vol.399, Part I, 2004,pp.278-282.

[5] T.C. Bailey, M.E. Colburn, B.J. Choi, A. Grot, J.G. Ekerdt, S.V. Sreenivasan,C.G. Willson, “Step and Flash Imprint Lithography: A Low-Pressure, RoomTemperature Nanoimprint Patterning Process”, in Alternative Lithography.

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[6] R. L.Burns, S. C.Johnson, G. M. Schmid, E. K. Kim, M. D. Dickey, J. Meiring, S.D. Burns, N. A. Stacey, C. G. Willson, “Mesoscale modeling for SFIL simulatingpolymerization kinetics and densification”, Proc. SPIE, 5374, 348-360 (2004).

[7] SLATEC Common Mathematical Library http://www.netlib.org/slatec

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