Organization of This Thesis

Chapter 1 Introduction

1.3 Organization of This Thesis

In this paper, the recently developed multiresolution time-domain (MRTD) method[12] is applied to fast and rigorous mask diffraction simulation, to overcome the limitations of FDTD. An outline of the MRTD formulation is described in Chapter 3. In addition, In chapter 4, using a 2D scattering example to allow the use of an extremely fine computational mesh to achieve good convergence.

1.3 Organization of This Thesis

The thesis includes four chapter. In chapter 1, we make an introduction to describe the background and polarization effects of the optical lithography and the role in the semiconductor industry that optical lithography is playing. In addition, we make a

brief to describe simple concept of the imaging process mathematically. Then, we describe the motivation of this thesis. Finally, we introduce PML and MRTD techniques respectively.

In chapter2, we will first specify a numerical scheme for solving the Maxwell’s equations, used by Yee.[13] Yee was one of the first to replace Maxwell’s equations by a set of finite difference equations. Then, we will illustrate PML technique for two-dimensional problems.

In chapter3, we first explain the most important properties of the multiresolution analysis. Then, we will only derive the S-MRTD scheme for a homogeneous medium Finally,in chapter4, we will use S-MRTD and PML schemes to rigorous simulate two-dimension phase-shifting mask structure by using matlab. Then, we will make a conclusion with simulation in the whole thesis.

Figure.1-1 Integrated circuit fabrication process

( ) a

( ) b

Figure.1-2 Frequently-encountered circuit geometries.

(a) lines, spaces, and contacts (b) periodic line-space pattern

( ) a

( ) b

Figure.1-3 (a) Optical diffraction blurs dark-bright transitions in the layout. (b) Photoresist nonlinearity can turn the rather sloppy image into vertical profiles.

Figure.1-4 Schematic of an exposure system.

Figure.1-5 High-NA vs low-NA

Figure.1-6 Aerial images of a 1:1 line-space pattern with a period of

0.7 /

NA

. The NA is 0.965.

Figure.1-7 Latent image contrast loss is less severe than aerial image degradation.

Figure.1-8 Immersion lithography rekindles the need to examine wafer polarization effects.

Figure.1-9 A simplified Model for an Imaging System

Chapter 2

Boundary Conditions for the Finite-Difference Time-Domain(FDTD) Method

2.1 The FDTD method and the Yee equations

The continuous form of the Maxwell equations for linear, isotropic, non-magnetic, nondispersive materials are written: of the frequency of the electromagnetic wave. For the application in hand, however,

they are assumed to be constant because of monochromatic excitation. Using Stokes' theorem, (1.ab) and (2) can be re-written in the weak form

l s

}

Where

F dl

∫ ⁱ

^{. and}

∫

^{F dS} ⁱ

represent, respectively, the line integral and surface integral of a variable

F

the field components are staggered and occupy distinct locations in space as shown in Fig. 2.1 The surface integral and line integral are thus evaluated on square surfaces as shown in Fig. 2.2. This discretization scheme[17] leads to three scalar equations in two-dimensional analysis for the transverse electric (TE) or the transverse magnetic (TM) polarization, but six equations in three-dimensional analysis for the field

components E E E H H_x, _y, _z, _x, _y,

and H_z

:

Whereα =

(2

ε σ−

) /(2

ε σ+

),

β =

(

/

) [2 /(2

⋅ ε σ+

)],

_and

(

t u x

/ )

γ =

. The superscript of the field variables stands for the time step (time = n⋅ ∆t), the subscript represents the direction of the field, and (i, j , k)

signifies the node position at

(

i x j y k z∆

,

∆

,

∆

) .

2.2 A Perfectly Matched Layer for the absorption of Electromagnetic Waves

2.2.1 The theory and derivation of the PML medium

(a) Definition of the PML medium

We will set the equation of a PML medium for two-dimensional problems,first in the TE(transverse electric) case. In Cartesian coordinates let us consider a problem that is without variation along z,with the electric field lying in the (x,y) plane(Fig.2.3) . The electromagnetic field involves three components only,Ex,Ey,Hz,and the Maxwell equations reduce to a set of three equations. In the most general case,which is a medium with an electric conductivity σ and a magnetic conductivity

σ ∗

,these equations can be written as

is satisfied,then the impendence of the medium (1) equals that of vacuum and no reflection occurs when a plan wave propagates normally across a vacuum-medium interface.

We will now define the PML medium in the TE case. The cornerstone of this definition is the break of the magnetic component Hz into two subcomponents which we will denote as H_zxandH_zy. In the TE case,a PML medium is defined as a medium in which the electromagnetic field has four

components,E E H_x

,

_zx

,

H_zy,connected through the four following equations:

(b) Propagation of a plane wave in a PML medium

Let us consider a wave whose electric field of magnitude

E

₀ forms an angle ϕ

with the y axis (Fig2.3). We will denote as

H

_zx₀and

H

_zy₀ the magnitudes of the magnetic components

H

_zxand

H

_zy. If a plane wave propagates in the PML

( ) constants. Since the magnitude

E

₀ is given,the set of Eq.(4)involves four unknown quantities to be determined,α β

, , H

_zxo

, H

_zyo. Enforcing

E E H

,

_zx

, H

_zy from(4) in the PML equations (3) yields the following set of equations connecting the four unknowns:

0 0 by means of writing the ratio (9) over (10),

0 two sets of (α ,β) of opposite signs for two opposite directions of propagation.

Choosing the positive sign we have

0 0

2 and the ratio Z of the electric magnitude over the magnetic one is

0 0 any frequency, and so the expression of the wave components (16) and of the impedance (19) become respectively

(c) Transmission of a Wave through PML-PML Interfaces

In this section,we will address the problem of a plane wave moving from a PML medium to another one. We will prove that in particular cases, with adequate sets of parameters σ σ_x, _x*,σ σ_y, _y*, the transmission is perfect and reflectionless at any incidence angle. These particular cases will be the basis of the perfectly matched

layer.

We first consider the case of two PML media separated by an interface normal to the x axis(fig2-4) . Let us denote θ₁ and θ₂ the angles of the incident and

transmitted electric fields E_i and E_t,with respect to the interface plane. In the case of (fig.2) we assume the interface to be infinite and incident wave to be plane.

First,both the reflected and transmitted waves must be plane too. Second, the ratios of these waves over the incident one must be without variation when moving on the interface. So, for any component ψ of the incident and transmitted fields, and for two points A and B of the interface, we can write

( ) ( )

Since (22) is true for any distance d, the exponential factors of (23) and (24) must be equal. So the relation

Let us now consider the incident, reflected, and transmitted electric and magnetic fields E E E H H H_i, _r, _t, _i, _r, _t(fig.2.4). First, continuity of the fields E_y and

zx zy

H +H lying in the interface yields the following set of equations:

1 1 2 the interface, with (2.2.11) and (2.2.14) we can write

1 1 1

As a consequence of the Snell-Descartes law (25),(26),the three exponentials on space of (29),(30),(31) are equal. So, reporting E_i….,H_t from (29),(30),(31) into (27),(28) Defining the reflection factor as the ratio of the electric components in the interface, that is, −E_rocosθ₁/E_iocosθ₁ ,and then solving the set (33),(34) for this ratio, the reflection factor r_p for the TE case is

2 2 1 1 And the reflection factor (36) is then

p 0

r = (39) The general frame of the PML technique is pointed out on Fig.2.5.

As a summary of this section we can now say that the reflection factor between two PML media whose conductivities satisfy (2) is null:

These are

the reflection factor is null at an interface normal to x lying between a vacuum and a (σ σ_x, _x^∗,0,0) matched medium or between a (0,0,σ σ_y, _y^∗) and a

(σ σ σ σ_x, _x^∗, _y, _y^∗) matched media.

the reflection factor is null at an interface normal to y lying between a vacuum and a (0,0,σ σ_y, ^∗_y) matched medium or between a (σ σ_x, _x^∗,0,0) and a

(σ σ σ σ_x, _x^∗, _y, _y^∗) matched media.

Let us consider, for instance, the upper-right part of a gridded computational domain (Fig2-6). With the usual FDTD notations, (3.b) and (3.c) yield the following equations that can be applied in the whole layer, except in the interface for E_y(see below), in the interface, the magnetic field has one component H_z on one side and two subcomponents H_zx,H_zy on the other. That is a result of using the Maxwell equations in the inner volume, but it has no physical significance. The finite

difference equations have to be modified. So, in the right side interface normal to x, (40) becomes

( ) /

2.3 Numerical Stability Condition

In the two-dimension case

2.4 Numerical Dispersion Relation

In the two-dimension TE(TM) case

ky are, respectively, the x- and y-components of the numerical wavevector, and w is the wave angular frequency. To quantitatively assess the dependence of numerical dispersion upon FD-TD grid discretization, we shall take as the case, assuming for simplicity square unit cells (x=y =) and wave

propagation at an angle α with respect to the positive x-axis(

kx =k cosα ;k_y =k sinα ). Then numerical dispersion relation () simplifies to

By applying the following Newton’s method iterative procedure:

So we can obtain the numerical wavevector

final

k

that the numerical phase velocity

See figure.2-7 graphs results obtained using this procedure that illustrate the

variation of the numerical phase velocity with propagation angle in a two-dimensional FDTD grid.

Fig2.1 Maxwell’s equations are solved over a cubic grid using the FDTD method.

The field components are staggered over the grid.

(a)

(b)

Fig2.2-(a)The electric field

E

_z(i,j,k) is calculated by summing up the magnetic field values of the four neighboring nodes. The magnetic field components are assumed to be constant along the line segments 1-2,2-3,3-4,and4-1,and the electric field

E

_z is assumed to be constant over the square surface bounded by 1-2-3-4.

(b) The magnetic field Hz(i+1/2,j+1/2,k+1/2) is calculated by summing up the electric field values of the four neighboring nodes. The electric field components are assumed to be constant along the line segments 1-2,2-3,3-4,and4-1,and the magnetic field Hz is assumed to be constant over the square surface bounded by 1-2-3-4.

Fig-2.3The transverse electric problem

Fig-2.4 Interface lying between two PML media.

Fig-2.5 The PML technique.

Fig-2.6 Upper-right part of the FDTD grid

wave propagation angle α

Figure.2-7 graphs results obtained using this procedure that illustrate the variation of the numerical phase velocity with propagation angle in a two-dimensional FDTD grid.

Chapter 3 Using Battle-Lemarie Scaling Function Based Multiresolution time-domain(MRTD) schemes

3.1 Fundamentals Of Multiresolution Analysis

In literature [1], [2], the use of scaling and wavelet functions as a complete set of basis functions is called multiresolution analysis.The most important properties of the scaling and wavelets functions defining a multiresolution analysis are now

summarized:

1. All scaling and wavelet functions are localized both in the frequency and the space/time domains.

2. The Fourier transform of a scaling function(Fig.3-1) is a low-pass

function(Fig.3-2), while the Fourier transform of the corresponding mother wavelet(Fig.3-3) is a bandpass function(Fig.3-4). This implies that, in a multiresolution expansion, scaling functions can be utilized to sample the low-frequency content of the field/signal, while wavelets sample the high-frequency content.

3. The scaling function of one resolution level, m, are orthonormal to the scaling functions of the same resolution level, but not the scaling functions of other resolution levels:

,

mv nv

φ φ

=

δ (3-1) This implies that, when using scaling functions in an expansion, we cannot mix different resolution levels.

4. The scaling function are orthogonal to all wavelets of any higher-order resolution level:

φ ψ_mn

,

_uv

= 0

m≤u (3-2) while the wavelet function of any resolution level are orthonormal to

all other wavelets at any resolution level:

,

uv mu nv

ψ ψ = δ δ

(3-3)

The above two properties allow us to mix scaling functions of one resolution level with wavelets of one or more resolution levels of the same or higher order, m.

5. Any scaling function of m resolution can be expanded in the form of wavelets of all lower resolution levels:

The S-MRTD and W-MRTD schemes[12] are derived using cubic spline Battle-Lemarie scaling and wavelet functions .

At first, for simplicity, we will only derive the S-MRTD scheme for a

homogeneous medium. The derivation is similar to that of Yee’s FDTD scheme which uses the method of moments[19] with pulse functions as expansion and test functions.

For the derivation of S-MRTD, the field components are represented by a series of scaling functions in space and pulse functions in time.

2-D (TE) S-MRTD Scheme: for a homogeneous medium with the permittivity ε and the permeability

u

₀ may be written in the form of three scalar Cartesian equations as

z x The electric and magnetic field components incorporated in these equations are expanded as following

space and time indices related to the space and time coordinates via

x = ∆ l x

, with the rectangular pulse function

1 1/ 2

whereφ

( ) x

represents the spline Battle-Lemarie scaling function [13], [14] depicted in Fig. 3-1. Assuming the Fourier transformation

( ) ( ) x e

^{j x}^λ

dx

Respectively, the closed-form expression of the scaling function in spectral domain is given by

with the low-pass spectral domain characteristics shown in Fig.3-2.

We insert the field expansions in Maxwell’s equations and sample the equations using pulse functions as test functions in time and scaling functions as test functions in space. For the sampling with respect to time, we need the following integrals[10]

' , ' For the sampling with respect to space, we use the orthogonality

relation for the scaling functions [17]

' , ' To calculate the integral corresponding to (15) for scaling functions, we

make use of the closed form expression of the scaling function in spectral domain.

According to Galerkin’s method [7], for complex basis functions, one has to choose the complex conjugant of the basis functions as test functions. We then obtain

' 1/ 2

< 0 are given by the symmetry relation a(-1-i)=-a(i). The Battle-Lemarie scaling function does not have compact but only exponential decaying support and thus, the coefficients a(i) for i > 8 are not zero. However, we found that these coefficients are negligible, affecting the accuracy of the field computation only for very low values of the wave vector. We therefore use the approximation

8 in order to obtain a MRTD scheme useful for practical applications.

For sake of simplicity,let us demonstrate the sampling of Maxwell’s

equations by using only scaling functions as expansion and testing functions. By applying Galerkin’s method,the system of (3),(4),(5) takes the form:

8 where l,m, and k are the discrete space and time indices. In fact, the total field at a particular space point for the S-MRTD scheme may be calculated from the field expansions, see (6) , by sampling them with delta test functions in space and time

domain. For example, the x-component of the total electric field

Due to the exponentially decaying support of the Battle-Lemarie scaling function (see Figure.3-l), only a few terms of this two-dimensional summation have to be considered.

3-3 Numerical Stability condition

In the two-dimension case and ∆ = ∆ = ∆x y

3-4 Numerical Dispersion

To calculate the numerical dispersion[14] of the S-MRTD scheme, plane monochromatic traveling-wave trial solutions are substituted in the discretized Maxwell’s equations. For example, theE E H_x, _y, _z components for the TE mode has is the wave angular frequency. Substituting the above expressions into (20), the

following numerical dispersion relation is obtained for the TE mode of the S-MRTD scheme after algebraic manipulation:

wave vector

k = (2 ) /

π λ_NUM , the dispersion relationship can be written as

Upon numerical solution of (23) by applying the bisection-Newton-Raphson hybrid technique for different values

n

_a,θ, and C, we obtain

k = (2 ) /

π λ_NUM

Figure.3-1 Cubic spline Battle-Lemarie scaling function in space domain.

Figure.3-2 Cubic spline Battle-Lemarie scaling function in spectral domain.

Figure.3-3 Cubic spline Battle-Lemarie wavelet function in space domain.

Figure.3-4 Cubic spline Battle-Lemarie wavelet function in spectral domain.

Table I Coefficients a(i)

Figure.3-5 Phase error of MRTD employing scaling functions only:

Graphs the variation of the numerical phase error in degrees per

wavelength as a function of the parameter, 1/s(1/C), for the case

n =

8

Chapter 4

Discussion and Conclusion

4.1 S-MRTD formulation

In the PML medium, we can obtain the equations(4.0) from chapter 2.

Upon applying Galerkin’s technique, the following is obtained by S-MRTD method

( ) ( )

In main grid(figure 4-1), we use the equations(4.5) from chapter 3.

4.2 Simulation results

In this thesis, we simulate a new contact hole 2D-PSM design(figure 4-7)[15], which will be compatible with dense contact-hole. When it is applied together with QUASAR OAI, we can enhance the printing resolution to achieve an excellent DOF(figure 4-8).When we proceeding to simulate this 3D-mask(figure 4-9), we can find the DOF degraded to 50 percent.(figure 4-10)

When we use our program by S-MRTD,we can make the actual reflection waves are short successfully.(see figure 4-2,3,4,5,6)

Accordingly, we can continue to simulate 2D-mask by the program.

4.3 Conclusion

The multiresolution time-domain method has been successfully applied to rigorous 2D mask diffraction simulation with PML. MRTD requires only about 3.5 to 3.9 nodes per wavelength to achieve a phase error of two degrees per wavelength or less, while FDTD requires 19 nodes per wavelength or more to achieve the same phase error(Table 1). As a result, MRTD is faster than the speed of FDTD for the same level of accuracy. Figure4- show the dispersion characteristics of FDTD and S-MRTD

Then,the development of the MRTD-PML absorber enhances the applicability of the MRTD technique to complex 2(3)-D open geometries while maintaining the high computational efficiency in terms of memory and execution time requirements.

After having completed the work related above, one way of further research were initiated: the implementation of the three-dimensional S-MRTD simulation in order to achieve a more detailed analysis for the wave-structure(MASK) interaction problem.

--- | | BACK PML | | ---

|L | / | R|

|E | (ib,jb) | I | |F | | G|

|T | | H|

| | MAIN GRID | T|

|P | | | |M | | P|

|L | (1,1) | M|

| |/ | L|

--- | | FRONT PML | | --- Figure 4-1 The PML regions are labeled as shown in the diagram

Figure4-2 MRTD simulation result

Figure 4-3 MRTD simulation result

Figure 4-4 MRTD simulation result

Figure 4-5 MRTD simulation result

Figure 4-6 MRTD simulation result

Figure 4-7 A new contact hole 2D-PSM design

best Defocus [痠]: -0.003, best Intensity Threshold: 0.404

DOF=0.6496 um

Figure 4-8 E-D curve of 100nm contact-hole by the 2D-PSM design

Figure 4-9 A new contact hole 3D-PSM design

SOLID-C ®

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

Defocus [痠]

0.35 0.40 0.45 0.50

Intensity Threshold

Width (CS) [痠]

0.090-0.110

0.3006 0.0315

best Defocus [痠]: 0.215, best Intensity Threshold: 0.450

DOF=0.3006um

Figure 4-10 E-D curve of 100nm contact-hole by the 3D-PSM design

∆ ∆

References

[1] S. G. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,”IEEE Trans. Pattem Anal. Machine Intell., vol.11, pp. 674-693, July 1989.

[2]B. Jawerth and W. Sweldens, “An overview of wavelet based multiresolution analyses,” SIAM Rev., vol. 36, no. 3, pp. 377-412, Sept.1994.

[3] J. D. Plummer, P. B. Griffin, and M. D. Deal, Silicon VLSI technology:

fundamentals, practice, and modeling, Pearson Education, first ed., 2000.

[4] B. J. Lin, Optical lithography, SPIE, first ed., 2002.

[5] M.Born and E.Wolf,principles of optics,7th ed.(1999)

[6] G. Furter, “High-resolution high-apertured objective." U. S. patent

#5880891, Apr. 1997.

[7] J.W.Goodman; "Introduction to Fourier Optics", Third ed (2005).

[8] D C.Cole,”Derivation and simulation of Higher Numerical Aperture Scalar Aerial Images”, Jpn. J. Appl. Phs. Vol. 31 (1992)pp.4110-4119

[9] M S. Yeung,”Extension of the Hopkins theory of partially coherent imaging to include thin-film interference effects”, SPIE Vol. 1927 Optical/Laser

Microlithography (1993)

[10]J. P. Be´renger, “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves”,J. Comput. Phys. 114, 185 (1994).

[11] Burm J. Lin,” Thin-film optimization strategy in high numerical aperture optical lithography part1: principles “, JM3. 4(4),2005

[12] M. Krumpholz and L. P. B. Katehi, "MRTD: New Time Domain Schemes Based on Multiresolution Analysis", IEEE Trans. Microwave

Theory Tech., vol. 44, pp. 555-561 (1996)

[13]K. S. Yee,”Numerical solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media”, IEEE Trans. Antenna. Propag. 14, 302 (1966).

[14] Emmanouil M. Tentzeris,” Stability and Dispersion Analysis of

Battle–Lemarie-Based MRTD Schemes”, IEEE transactions on microwave , vol. 47, no. 7, 1999

[15] Chang, Chung-Hsing “Novel contact hole reticle design for enhanced lithography process window in IC manufacturing” , Proc. of SPIE. vol. SPIE-5645, pp.32-43.

2005

[16] A.K.Wong,”Wave-optical considerations in photolithography”,Proceedings Of SPIE vol.5182(2003)

[17]TAFLOVE,Computational Electrodynamics: The Finite-Difference Time-Domain Method.Boston:Artech House,1995.

[18] TAFLOVE,Advances in Computational Electrodynamics:The Finite-Difference Time-Domain Method.Boston,London:Artech House,1998

[19] R. F. Harrington, Field Computation by Moment Methods. Malabar,FL: Krieger , 1982.

[20] A.K.Wong , A R.Neureuther,”Polarization effects in mask Transmission”, SPIE vol.1674 Optical/Laser Microlithography(1992)

個人簡歷姓名：曾建彰

性別：男

出生年月日：民國 71 年 11 月 14 日住址：高雄市苓雅區正大路 120 之 5 號

學歷：

國立高雄大學電機工程系學士 (91.9-94.6) 國立交通大學電子研究所碩士 (94.9-96.7)

碩士論文題目：

二維完美匹配層應用於多重解析法之光學微影模擬

Two-Dimensional Perfectly Matched Layer Applied to Multiresolution Time-Domain Method for Optical lithography simulation

在文檔中二維完美匹配層應用於多重解析法之光學微影模擬 (頁 16-0)

Chapter 1 Introduction

1.3 Organization of This Thesis

1.3 Organization of This Thesis

( ) a

( ) b

( ) a

( ) b

0.7 /

NA

Chapter 2

Boundary Conditions for the Finite-Difference Time-Domain(FDTD) Method

2.1 The FDTD method and the Yee equations

}

F dl

∫ i

∫

F dS i

F

:

(2



) /(2



),

( 

/ 

) [2 /(2



)],

(

/ )

 

(

,

,

) .

2.2 A Perfectly Matched Layer for the absorption of Electromagnetic Waves

2.2.1 The theory and derivation of the PML medium

(a) Definition of the PML medium

σ ∗

,

,

,

(b) Propagation of a plane wave in a PML medium

E

H

H

H

H

E

, , H

, H

E E H

,

,

, H

(c) Transmission of a Wave through PML-PML Interfaces

2.3 Numerical Stability Condition

2.4 Numerical Dispersion Relation

k

(a)

(b)

E

E

Chapter 3

Using Battle-Lemarie Scaling Function Based Multiresolution time-domain(MRTD) schemes

3.1 Fundamentals Of Multiresolution Analysis

,

=

,

= 0

,

ψ ψ = δ δ

u

x = ∆ l x

1 1/ 2

( ) x

( ) ( ) x e

dx

∫ ⁱ

^{F dS} ⁱ

(

/

個人簡歷姓名：曾建彰

出生年月日：民國 71 年 11 月 14 日住址：高雄市苓雅區正大路 120 之 5 號