Step-width Adjustment and Sidewall Control in Electron-beam Lithography
by
Pengcheng Li
A thesis submitted to the Graduate Faculty of
Auburn University
in partial fulflllment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
December 13, 2010
Keywords: proximity efiect correction, dose control, step-width adjustment, sidewall control
Copyright 2010 by Pengcheng Li
Approved by
Soo-Young Lee, Professor of Electrical and Computer Engineering
Stanley J. Reeves, Professor of Electrical and Computer Engineering
Bogdan M. Wilamowski, Professor of Electrical and Computer Engineering
Abstract
Two-dimensional (2-D) patterns and three-dimensional (3-D) structures increasingly
flnd applications in various devices such as difiractive optical elements, photonic element,
microelectromechanical systems (MEMS) etc. and are often fabricated by e-beam lithog-
raphy. Their performance is known to be highly sensitive to their dimensions. Therefore,
it is critical to achieve high dimensional accuracy for the desired characteristics. However,
as the feature size decreases down to nanoscale, the non-ideal exposure distribution due to
electron scattering can make dimensions of the fabricated features in a device substantially
difierent from the target dimensions. In this thesis, the issue of controlling the dimensions
of the features transferred onto the resist layer is addressed for the staircase structures and
the line patterns. The remaining resist proflle estimated from the 3-D exposure distribution
is employed in the optimization procedure in order to obtain realistic results. The results
from the experiments and extensive simulation study are analyzed.
ii
Acknowledgments
I would like to thank my advisor Dr. Soo-Young Lee for all his support, detailed
guidance and helpful suggestions throughout the entire development of PYRAMID for three-
dimensional lithography without which this thesis would not have been possible. I would also
like to thank the other members of my committee, Dr. Stanley J. Reeves and Dr. Bogdan M.
Wilamowski for their support and help. Thanks are also due to the National NanoFab Center
for experiment and characterization services, and Samsung Electronics Co. for funding this
research, and to Auburn University and Department of ECE for their flnancial support.
Also, I would like to thank Mr. Qing Dai for helping me get familiar with the PYRAMID
programs. Last but not least, I would like to thank my family, my friends and all other
people who have given me help during my MS study.
To my loved family members and heavenly abodes, without whose support I would not
be able to be where I am now.
iii
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Estimation of Remaining Resist Proflle . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Electron Beam Lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Point Spread Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 3-D Exposure Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Developing Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Width Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.6 Resist Development Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.6.1 Resist Development Algorithm of Discrete-level Structures . . . . . . 9
2.6.2 Resist Development Algorithm of Continuous Structures . . . . . . . 10
3 Adjustment of Step-Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.1 Derivation of Developing Rate . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Estimation of Step-Width Deviation . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Step-Width Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
iv
3.4.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Controlling Sidewall of Resist Proflle . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1 3-D Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1.1 Developing Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1.2 Simulation of Resist Development . . . . . . . . . . . . . . . . . . . . 26
4.2 Sidewall Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4 Trend Analysis of Sidewall Control . . . . . . . . . . . . . . . . . . . . . . . 38
4.4.1 The Efiect of Developing Time on Dose Distribution . . . . . . . . . 38
4.4.2 The Efiect of Average Dose on Dose Distribution . . . . . . . . . . . 41
4.4.3 The Efiect of Developing Time on Sidewall Shape . . . . . . . . . . . 42
4.4.4 The Efiect of Dose Distribution on Sidewall Shape . . . . . . . . . . . 42
5 Concluding Remarks and Future Study . . . . . . . . . . . . . . . . . . . . . . . 45
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
v
List of Figures
2.1 Illustration of the e-beam grayscale lithographic process. . . . . . . . . . . . . 4
2.2 A PSF for the substrate system of 500 nm PMMA on Si with the beam energy
of 50 KeV : (a) the top, middle and bottom layers, and (b) all layers . . . . . . 5
2.3 (a) A line pattern transferred onto resist and (b) the cross section of the remaining
resist proflle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Cross section of the remaining resist for a line feature: (a) overcut and (b) un-
dercut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Estimation of step-edge deviation where ?ti is the time taken for the half of the
step-height to be developed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 Step-edge deviations in a staircase structure . . . . . . . . . . . . . . . . . . . . 11
2.7 The remaining resist proflle of discrete-level structure. . . . . . . . . . . . . . . 11
2.8 The remaining resist proflle of a line pattern estimated from (a) cell removal and
(b) new method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1 (a) A nine-step staircase structure and (b) its cross section where the solid and
dashed lines are target and actual proflles, respectively. . . . . . . . . . . . . . 14
3.2 Exposure-developing rate curve for MIBK:IPA = 1:1 (lines with symbols) with
developing time of 30 sec. The circles are experimental data to which a part of
Gaussian curve is fltted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Step-width adjustment scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
vi
3.4 (a) Step-edge deviation: f?Wig = f0, 45, 70, 90, 125, 125, 90, 70, 45, 0 nmg
Step-width deviation: f?Wig = f45, 25, 20, 35, 250, 35, 20, 25, 45 nmg (b) Step-
edge deviation: f?Wig = f0, 0, 0, 0, 0, 0, 0, 0, 0, 0 nmg Step-width deviation:
f?Wig = f0, 0, 0, 0, 0, 0, 0, 0, 0 nmg Remaining resist proflles obtained (through
simulation) for the staircase structure (a) before adjustment and (b) after step-
width and dose adjustments where target step-width: 1.5 ?m, target step-height:
200 nm, resist thickness: 1000 nm on Si (50 KeV). A deviation less than 1 nm
is rounded to zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 (a) Step-edge deviation: f?Wig = f0, 15, 25, 35, 60, 60, 35, 25, 15, 0 nmg
Step-width deviation: f?Wig = f15, 10, 10, 25, 120, 25, 10, 10, 15 nmg (b) Step-
edge deviation: f?Wig = f0, 0, 0, 0, 0, 0, 0, 0, 0, 0 nmg Step-width deviation:
f?Wig = f0, 0, 0, 0, 0, 0, 0, 0, 0 nmg Remaining resist proflles obtained (through
simulation) for the staircase structure (a) before adjustment and (b) after step-
width and dose adjustments where target step-width: 1.0 ?m, target step-height:
100 nm, resist thickness: 500 nm on Si (50 KeV). A deviation less than 1 nm is
rounded to zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.6 Experimental results (a) before step-width adjustment and (b) after step-width
adjustment where target step-width: 1.5 ?m, target step-height: 200 nm, resist
thickness: 1000 nm PMMA on Si (50 KeV). . . . . . . . . . . . . . . . . . . . . 23
3.7 Step-width deviations in simulation and experimental results (a) before step-
width adjustment and (b) after step-width adjustment for 1000 nm PMMA (50
KeV). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1 An illustration of sidewall shape speciflcation in the cross section: rxi and pxi
are the target and actual widths of line in the ith layer of resist, respectively.
The cost function is deflned as C = maxi(jrxi ?pxij). . . . . . . . . . . . . . . 27
4.2 Dose distribution and the corresponding sidewall shape: (a) a uniform dose dis-
tribution and (b) the spatially-controlled dose distribution. . . . . . . . . . . . 28
vii
4.3 The  owchart of SA (simulated annealing) process . . . . . . . . . . . . . . . . 29
4.4 During the SA process, the doses of all regions of a line feature are adjusted. The
solid line and dashed lines represent the dose distribution before and after dose
adjustment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.5 Dose step range vs temperature in the SA process. . . . . . . . . . . . . . . . . 31
4.6 Three dose distributions of Distribution-A, Distribution-B and Distribution-C . 32
4.7 The remaining resist proflles (sidewall shapes) of Distribution-A, Distribution-B
and Distribution-C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.8 Simulation results with the same average dose 500 ?C=cm2 (a) simulation result
for Distribution-A (b)simulationresult for Distribution-B (c)simulationresult for
Distribution-C where developing time: 40 sec, MIBK:IPA=1:2, 300 nm PMMA
on Si (50 KeV). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.9 Simulation results with the same average dose 525 ?C=cm2 (a) simulation result
for Distribution-A (b)simulationresult for Distribution-B (c)simulationresult for
Distribution-C where developing time: 40 sec, MIBK:IPA=1:2, 300 nm PMMA
on Si (50 KeV). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
viii
List of Tables
3.1 Experimental result for a nine-step symmetric staircase structure fabricated on
1000 nm PMMA on Si (50 KeV): dose, exposure, and depth. . . . . . . . . . . 15
4.1 Efiects of the dose distribution on the sidewall shape with the total (average)
dose and developing time flxed. . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 The developing time required to achieve the same (equivalent) sidewall shape
with the total (average) dose flxed. . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 The total (average) dose required to achieve the same (equivalent) sidewall shape
with the developing time flxed. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4 Comparison of dose distributions with the same target resist proflle (vertical
sidewall) for line width of 100 nm and the same average dose (resist thickness:
300 nm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.5 Comparison of dose distributions with the same target resist proflle (vertical
sidewall) for line width of 100 nm and the same average dose (resist thickness:
100 nm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.6 Comparison of dose distributions with the same target resist proflle (vertical
sidewall) for line width of 100 nm and the same average dose (resist thickness:
500 nm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.7 Comparison of dose distributions with the same target resist proflle (undercut
sidewall) for line width of 100 nm and the same average dose (resist thickness:
300 nm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.8 Comparison of dose distributions with the same target resist proflle (vertical
sidewall) for line width of 100 nm and the same developing time (resist thickness:
300 nm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.9 Comparison of dose distributions with the same target resist proflle (vertical
sidewall) for line width of 100 nm and the same developing time (resist thickness:
100 nm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.10 Comparison of dose distributions with the same target resist proflle (vertical
sidewall) for line width of 100 nm and the same developing time (resist thickness:
500 nm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
ix
4.11 Comparison of dose distributions with the same target resist proflle (undercut
sidewall) for line width of 100 nm and the same developing time (resist thickness:
300 nm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.12 The efiect of developing time on sidewall shape with the average dose (560.0
?C=cm2) and dose distribution flxed (resist thickness: 300 nm). . . . . . . . . 43
4.13 The efiect of developing time on sidewall shape with the average dose (650.0
?C=cm2) and dose distribution flxed (resist thickness: 500 nm). . . . . . . . . 44
4.14 Comparison of dose distributions required to achieve a target sidewall shape for
difierent resist thicknesses. For each given developing time, the average dose is
minimized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
x
Chapter 1
Introduction
Miniaturization and performance improvements are the central concerns in modern fab-
rication technology, and electron-beam (e-beam) lithography is one of the key technologies to
fabricate devices at nanometer scale due to its very short wavelength [1]. Two-dimensional
and three-dimensional patterns are transferred onto a resist layer via e-beam lithographic
process in various applications, e.g., discrete devices [2], photomasks [3], molds for imprint
lithography [4], etc. However, proximity efiect (or non-ideal exposure distribution) due to
the forward and backward-scattering of electrons results in undesirable blurring of the trans-
ferred pattern. The degree of proximity efiect mainly depends on the beam accelerating
voltage, resist thickness, substrate material and beam diameter [5].
As the feature size is reduced well below a micron and the circuit density continues to
increase, the relative variation of feature dimensions, due to the proximity efiect, becomes
larger. Hence, it is crucial to have a practical scheme to minimize the dimensional deviation
from the target pattern. In most schemes, an empirical approach which relies on the experi-
mentally determined relationship between the e-beam dose or exposure and remaining resist
thickness is taken. However, such an approach takes into account neither the resist devel-
opment process nor possible variation of feature size, thus possibly resulting in substantial
dimensional errors [6?9]. Therefore, in order to have an accurate control of the remaining
resist proflle, an analytic model which considers the resist development processes, in addi-
tion to the exposure distribution, is required. In this study, such a model is employed for
step-width adjustment in staircase structures and controlling sidewall shape in line patterns.
1
1.1 Previous Work
Numerous researches on proximity efiect correction and the related issues have been
conducted by many researchers for many years [10 ? 20]. The fundamental di?culties in
proximity efiect correction are the large size of data involved and the constraint of non-
negative solution (dose). The two critical issues are the accuracy and speed of correction.
Most of the correction schemes adopted either dose or shape modiflcation to achieve a desir-
able exposure distribution. In general, the accuracy and complexity of a correction scheme
depend on the model which describes the e-beam lithographic process, in addition to the
correction scheme itself. Many models ignore the forward-scattering of electrons and the
exposure variation along the depth dimension of resist.
PYRAMID [21 ? 35], a hierarchical rule-based approach toward proximity efiect cor-
rection, was introduced for fast and accurate correction. It has been demonstrated that
PYRAMID can handle the feature size of nanoscale. Its correction hierarchy is  exible such
that various correction models can be implemented. The PYRAMID implementations in-
clude the shape modiflcation [27], the dose modiflcation [36], the heterogeneous substrate
correction [37], the grayscale correction [38], the non-rectangular feature correction [39], etc..
More recently, the PYRAMID was further developed by employing a 3-D model where the
exposure variation along the depth dimension is considered. The difierence between the 2-D
and 3-D models, in terms of their efiects on correction results, was analyzed [40]. Correction
based on the 3-D model was considered for discrete-level grayscale structures and binary
patterns. In particular, controlling the step depth in staircase structures was investigated
via simulation and experiment [41]. Also, controllability of the sidewall shape in line patterns
was studied through simulation [42].
1.2 Motivation and Objectives
It was noticed in the fabrication of discrete-level structures such as a staircase that
the actual step-width can be substantially deviated from the target width. Therefore, it is
necessary to control the width as well as the height of each step. Also, in the case of line
pattern, a difierent sidewall shape of line may be preferred in a difierent application. Hence,
2
it is desirable to have a certain degree of control over the sidewall shape. The problem of
sidewall control can be viewed as a generalization of the step-width control in a discrete-level
structure where the number of levels (steps) is large. The objectives of this study are to
develop a step-width adjustment scheme and further reflne the sidewall control scheme based
on the recent results from the PYRAMID project.
1.3 Organization of the Thesis
This thesis is organized as follows:
? Chapter 2 introduces the algorithms for estimating remaining resist proflle based on
the 3-D exposure model.
? Chapter 3 describes the step-width adjustment scheme for staircase structures, which
employs both dose and shape modiflcations.
? Chapter 4 includes the results from analyzing the relationship among the total dose,
spatial distribution of dose, developing time and sidewall shape, and implementing the SA
(simulated annealing) for dose optimization
? Chapter 5 presents conclusions and suggestions for the future work.
3
Chapter 2
Estimation of Remaining Resist Proflle
In this chapter, the 3-D model of exposure distribution (therefore rate distribution) and
resist development simulation are brie y reviewed [47], following an introduction of e-beam
lithography.
2.1 Electron Beam Lithography
E-beam lithography is the process of transferring the circuit patterns from photo litho-
graphic mask to the resist using a focused beam of electrons. The primary advantage of
e-beam lithography is that it ofiers high patterning resolution and versatile pattern forma-
tion. Thus e-beam lithography is the most commonly used technique for nanolithography.
Figure 2.1: Illustration of the e-beam grayscale lithographic process.
As illustrated in Figure 2.1, a resist is exposed by e-beam, depositing energy in the
resist, and the amount of energy at each location depends on how electrons are scattered.
4
Then a solvent developer is used to selectively remove either exposed (positive photoresist)
or unexposed regions (negative photoresist). For the positive photoresist, the resist is re-
moved where the energy deposited is higher than a certain threshold exposure value. After
development process, the remaining resist proflle serves as a mask to selectively etch the
substrate material, transferring the circuit patterns onto the substrate if necessary.
2.2 Point Spread Function
For high-quality proximity efiect correction and estimation of resist proflle, the accurate
knowledge of point spread function (PSF) is required in e-beam lithography. A PSF shows
how the electron energy is distributed throughout the resist when a single point is exposed.
As shown in Figure 2.2 (a), the PSF is a function of the distance from the exposed point
and radically symmetric in three dimensions. In general, the PSF depends on the resist
thickness, beam energy, beam diameter, substrate composition, etc., and is independent on
the dose given to the point. For homogeneous substrate, the PSF does not vary with the
position of the point exposed.
(a) (b)
Figure 2.2: A PSF for the substrate system of 500 nm PMMA on Si with the beam energy
of 50 KeV : (a) the top, middle and bottom layers, and (b) all layers
Theoretical modeling such as a double-Gaussian function or a Monte Carlo simulation
[43 ? 44] is used to get the PSF. It can be seen from Figure 2.2 (a) and (b) that a PSF
5
can be decomposed into two components, the local component due to electron?s forward-
scattering and the global component due to electron?s backward-scattering. Within the local
component, the PSF has large magnitude and is very sharp, but it varies rapidly. While the
magnitude of the global component is orders of magnitude lower than that in the forward-
scattering range and varies slowly.
2.3 3-D Exposure Model
E-beam lithographic process can be assumed to be linear and space invariant for uniform
substrates. Therefore, the exposure deposited in the resist can be estimated by the convo-
lution between the circuit pattern (dose distribution) and a PSF. In 3-D exposure model, a
3-D PSF is used, and thus the depth-dependent proximity efiect is considered.
Consider an X-Y plane which corresponds to the top surface of the resist layer, as shown
in Figure 2.3 (a). Let d(x;y;0), e(x;y;z), and psf(x;y;z) represent the e-beam dose to the
point (x;y;0) at the surface of the resist, the exposure at the point (x;y;z) in the resist, and
the PSF, respectively. Then the 3-D exposure distribution can be computed as follows:
e(x;y;z) =
Z Z
d(x?x0;y ?y0;0)psf(x0;y0;z)dx0dy0 (2.1)
In this study, exposure is computed by an accurate and e?cient two-level procedure
implemented in the PYRAMID software [24] which is decomposed into local exposure and
global exposure. Local exposure is the exposure contributed by the features close to the
exposed point under consideration and global exposure is the sum of exposure contributions
by features far away from the point at which exposure is calculated. When the circuit pattern
is su?ciently long (in the Y-dimension in Figure 2.3 (a)), exposure can be assumed not to
vary along the Y-dimension in most of the structure. In such a case, consideration of only
a cross section in the middle of the pattern shown in Figure 2.3 (b) is su?cient. Hence, in
the remainder of this thesis, the Y-dimension is not taken into account, i.e., only the cross
section in the X-Z plane will be considered.
6
(a)
(b)
Figure 2.3: (a) A line pattern transferred ontoresist and (b) the cross section of the remaining
resist proflle.
Though the 3-D exposure model estimates how electron energy is distributed throughout
the resist, it does not depict the remaining resist proflle explicitly after development. There-
fore, the resist development process simulation should be taken into account for proximity
efiect correction in order to get more realistic and accurate correction results.
7
2.4 Developing Rate
To simulate the remaining resist proflle after development, the resist developing rate
matrix r(x;z) (nm/min) at point (x;z) is transformed from the exposure matrix e(x;z)
(eV/?m3). The relationship between exposure and resist developing rate is known to be
nonlinear [46]. For high accuracy of proximity efiect correction, the relationship has been
derived from the experimental results. The exposure-to-rate conversion formula depends on
developing time and the resist-solvent combination. As exposure increases, developing rate
increases more than linearly, i.e., the increase is slow in the beginning and then faster. But,
when exposure exceeds a certain value, developing rate tends to saturate (though this region
is not utilized in this study). Such behavior of developing rate may be modeled by a curve or
a part of curve with an in ection point, including the 3rd polynomial and Gaussian curves.
2.5 Width Variation
The width of a feature such as a single line or each step of a staircase structure may
vary along the resist depth as illustrated in Figure 2.4. Let W(z), Wt, Wm, and Wb denote
the width of the feature which is a function of resist depth z, the widths of a feature at the
top, middle and bottom layers of the resist, respectively. Note that Wt > Wm > Wb and
Wt < Wm < Wb indicate overcut and undercut sidewalls, thus the order of feature width
W(z) specifles the type of sidewall. Due to the lateral development of the resist and spatial
distribution of dose, the actual feature width deviates from the target one, even causing
the sidewall shape to be changed. For example, the actual resist proflle is overcut, though
the desired one is undercut or vertical sidewall. Therefore, the proximity efiect correction
method is carried out to minimize the deviation between the actual and target widths of a
feature.
8
Figure 2.4: Cross section of the remaining resist for a line feature: (a) overcut and (b)
undercut
2.6 Resist Development Algorithm
With the advancement of MEMS fabrication techniques and the increasing complexity
of IC design, accurate and e?cient simulation processes such as resist development, are
greatly needed, in order to reduce development cost and fabrication experiments. One of
the di?culties in resist development simulation is the resist development algorithm. Several
kinds of algorithms have been proposed: the ray tracing algorithm, the string algorithm, and
the cell removal algorithm. The former two algorithms sometimes causes fatal errors such as
looping of rays or strings. While the cell removal algorithm is absolutely stable and robust
[47] but it costs considerably computational time. Also, the method to estimate the resist
proflle depends on the patterns under consideration.
2.6.1 Resist Development Algorithm of Discrete-level Structures
For discrete-level structures, such as staircase structures in Figure 3.1(a), one way to
estimate the resist proflle is to rely on a full-scale resist development simulation such as the
cell removal method which is too time-consuming to be practical especially in an iterative
procedure. Also, for step-width adjustment, only step-widths, rather than a complete resist
proflle, need to be estimated. A practical and yet su?ciently accurate method has been
9
developed for step-width estimation. Using e(x;z) obtained by simulation and the flnal
conversion formula, r(x;z) is computed. The slope of step-edge is also afiected by the
proximity efiect and is not usually vertical. However, for simplicity, the width of a step
is measured at the middle level between two adjacent steps, as illustrated in Figure 2.5.
For each step in the staircase, the time duration, ?ti , during which the second half of
the flnal step-height is developed, is derived based on r(x;z) (refer to Figure 2.5). Then the
location of the step-edge in the developed resist proflle is estimated by computing the amount
of lateral development during ?ti, from which the step-width deviation is obtained. The
deviation of step-edge with respect to the target location is denoted by ?Wi, as illustrated
in Figure 2.6, and the deviation of step-width compared to the target width is represented
by ?Wi = j?Wi ??Wi+1j for i = 1, 2, 3, and 4 (2?Wi for i = 5). Finally, the actual step-
width W0i of step i can be calculated by W0i = Wi??Wi where Wi is the target step-width of
step i. Figure 2.7 is one application of resist proflle estimation using the proposed method
for discrete-level structures, thus this method is developed to adjust step-width of staircase
structures.
Figure 2.5: Estimation of step-edge deviation where ?ti is the time taken for the half of the
step-height to be developed.
2.6.2 Resist Development Algorithm of Continuous Structures
For continuous features, such as the line pattern, the full-scale resist development algo-
rithm is still required to depict the resist proflle. A simplifled version of the resist develop-
ment process model, PEACE [47], is employed to estimate the remaining resist proflle after
10
Figure 2.6: Step-edge deviations in a staircase structure
X (?m)Resist depth (nm) 1.5 3 4.5 6 7.5 9 10.5 12 13.5
200
400
600
800
1,000
Figure 2.7: The remaining resist proflle of discrete-level structure.
development. The cell removal method is based on time evolution of the resist front, and the
amount of each cell to be etched in each step depends on the minimum dissolution time dT
and its neighboring cells in contact with the developer. The dT, the minimum dissolution
time to fully develop a cell in the current surface front, is computed and then the status
of other cells are updated for the elapsed time dT. When a cell is removed, its neighboring
cells start to be developed. Note that the dT is estimated from not only the cell under
consideration, but also its neighbors. By tracking and updating the status of all cells, the
cell removal method is able to simulate the resist development process. Thus, the cells are
removed in the order that the development proceeds.
The computationally-intensive nature of the cell removal method makes the proxim-
ity efiect correction procedure extremely time-consuming since the development simulation
11
needs to be carried out many times through iterations. Also, it often results in rough re-
sist proflles. With the assumption that a feature varies only in one dimension such as a
long line, a new simulation method was recently developed in our group, which is orders of
magnitude faster than the cell removal method and generates smooth proflles. The overall
shapes of proflles obtained by the new method are equivalent to the respective proflles by
the cell removal method. It flrst considers only vertical development and subsequently all
possible developing paths consisting of lateral development following vertical development.
The resist proflles obtained by the cell removal and new method are shown in Figure 2.8.
X (nm)
Resist Depth (
nm)
10020030040050050 150250350450
50100
150200
250300
X (nm)
Resist Depth (
nm)
0 50100150200250300350400450500
50100
150200
250300
(a) (b)
Figure 2.8: The remaining resist proflle of a line pattern estimated from (a) cell removal and
(b) new method.
12
Chapter 3
Adjustment of Step-Width
The issue of improving the dimensional accuracy of 3-D structures fabricated by e-
beam grayscale lithography is addressed for structures with discrete levels. Speciflcally, a
step-width adjustment scheme is developed for accurately controlling the (remaining) resist
proflles of staircase structures. The goal of the scheme is to minimize the step-width error
in the resist proflle. The scheme is based on 3-D exposure and resist development models
[41]. An exposure-to-developing rate conversion formula is derived based on experimental
results. Then, a resist development scheme with the rate conversion formula is employed to
estimate the resist proflle from which the amount of adjustment in step-width is determined.
Performance of the step-width adjustment scheme has been analyzed through computer
simulation and also experiments. It has been shown that this practical scheme is efiective in
reducing the step-width error.
3.1 Model
In Figure 3.1(a), a symmetric nine-step staircase structure is illustrated and the cross
section in the middle of the staircase is shown in Figure 3.1(b). As shown in Figure 3.1(b),
step i refers to the ith step from the left.
3.1.1 Derivation of Developing Rate
Let the relationship be represented by a non-linear function F[ ] to be referred to
as (exposure-to-developing rate) conversion formula. Then, r(x;z) is given by r(x;z) =
F[e(x;z)]. Letp(x)denotetheresistproflle, i.e, thedepthmeasuredfromtheinitialsurfaceof
the resist downward as shown in Figure 3.1(b). When the step-width is wide, the developing
process at the center of the step progresses mainly in the vertical direction such that the
lateral development may be ignored. Then, for the center of each step, the depth p(x) can
13
(a)
(b)
Figure 3.1: (a) A nine-step staircase structure and (b) its cross section where the solid and
dashed lines are target and actual proflles, respectively.
be related to the developing rate in the contiguous domain, as in Equation 3.1 where T is
the developing time.
Z p(x)
0
dz
r(x;z) = T (3.1)
In the experimental result of a staircase structure, only the discrete depth information
is available, i.e., one depth for each step (refer to Table 3.1). Let pi denote the depth of step i
measured at the center of the step in an experimental result and qi the target depth of step i.
The depth error of step i is represented by ?di = jpi?qij. Also, the cross section is partitioned
into blocks, as shown in Figure 3.1(b). Exposure is considered to be homogeneous within
each block and eij denotes the exposure in the jth block of step i.
14
Developer Step Dose (?C/cm2) Exposure (eV/?m3) Depth pi (nm)
1 216 1:7286?1010 182
2 296 2:0983?1010 417
MIBK:IBA = 1:1 3 334 2:2489?1010 626
4 364 2:3751?1010 852
5 396 2:5229?1010 1000
Table 3.1: Experimental result for a nine-step symmetric staircase structure fabricated on
1000 nm PMMA on Si (50 KeV): dose, exposure, and depth.
Derivation of the exposure-to-rate conversion formula is carried out in two phases. In
the flrst phase, the depth-dependent exposure or rate variation is not considered by using the
average value of eij for each step, which is denoted by ei. Accordingly, the average developing
rate ri for each step is deflned. From experiments, pi is obtained and ri is estimated to be piT
for step i. Then, the set of sample points, fei, rig, is fltted to a curve to derive the conversion
formula. The behavior of developing rate may be modeled by a Gaussian curve. As shown in
Figure 3.2, a part of the left half of a Gaussian function ri = a?exp(?(ei?bc )2) is employed
in order to minimize the number of coe?cients to be determined in curve fltting. Note that
a third-order polynomial involves four coe?cients. One problem of the flrst phase is that
it does not take the depth-dependent exposure (and therefore rate) variation into account,
which would cause a signiflcant error in estimating depth and width of a step.
In the second phase, the conversion formula obtained in the flrst phase is used as an
initial solution for an iterative reflning procedure. In each iteration, the block-wise exposure
distribution eij is used to estimate the depth of step i, to be denoted by p0i, based on the
current conversion formula. Then, the coe?cients a, b, and c, in the conversion formula are
adjusted such that the error Pjpi ? p0ij is minimized through an exhaustive search. From
several sets of experimental data, the following conversion formula was obtained:
r(x;z) = 4024:2?e?(e(x;z) ?3:4 ? 10
10
1:0968 ? 1010 )
2 (3.2)
where e(x;z) is in eV/?m3 and r(x;z) is in nm/min.
15
1.5 2 2.5 3 3.5 4 4.5 5x 1010
0.5
1
1.5
2
2.5
3
3.5
Exposure (eV/?m3)
Developing rate (
?m/min)
Figure 3.2: Exposure-developing rate curve for MIBK:IPA = 1:1 (lines with symbols) with
developing time of 30 sec. The circles are experimental data to which a part of Gaussian
curve is fltted.
The average percent error in curve fltting is 4.41% and the range of exposure used in
the adjustment scheme is from 0 to 3:4?1010 eV/?m3.
16
3.2 Estimation of Step-Width Deviation
Due to the isotropic process of resist development, the vertical wall between adjacent
steps in a staircase structure is developed laterally, causing the step-width to be difierent
from the target width. In Figure 3.4(a), a simulated resist proflle of staircase structure is
provided where it can be seen that the deviation of center step-width is as large as 16%. For
step-width adjustment, the step-width deviation flrst needs to be estimated as described in
Section 2.6. It should be noted that in the case of the staircase structure considered in this
study, step-width adjustment is equivalent to step-edge adjustment.
3.3 Step-Width Adjustment
A practical scheme for adjusting step-widths has been developed that minimizes the
computational requirement by avoiding a complete resist development simulation. As shown
in Figure 3.3, the scheme proceeds as follows.
Figure 3.3: Step-width adjustment scheme
Step 1: The 2-D exposure distribution is computed by the PYRAMID software[24] and the
initial developing rate formula is derived, based on the experimental result (depth of
each step).
17
Step 2: The conversion formula is reflned iteratively, using the 3-D exposure distribution com-
puted by the PYRAMID software (refer to Section 2.3).
Step 3: The step-edge deviation, ?Wi, is estimated for each step, given a developing time,
using the method described in Section 2.6.
Step 4: The amount of step-edge adjustment, ?W0i, to compensate for ?Wi, is determined
through an iterative procedure. In each iteration, the edge location (equivalent width)
of each step (step i) is adjusted by ?Wi before recomputing the exposure distribution
from which ?Wi is estimated, as in Step 3. This is repeated until the adjustment
of the edge location in an iteration is less than a half of pixel. The flnal step-edge
locations (or widths) obtained through the iterative procedure are further adjusted to
compensate for the minor overestimation by simulation, i.e., ?W0i ? c?dWi, where
dWi is the difierence between the initial and flnal step-edge locations and the c is a
constant less than 1.
Step 5: In many cases, one may not achieve the target widths with high precision due to the
fact that the step-width can be adjusted only by whole pixels. Also, an adjustment of
step-width also changes the exposure distribution, especially in the neighboring areas.
Hence, in order to have flner control in adjusting step-width and also to compensate
for the exposure change due to step-width adjustment, the dose adjustment may be
carried out along with the width adjustment. The dose adjustment is carried out by
the grayscale PYRAMID software [41].
3.4 Results and Discussion
3.4.1 Simulation Results
The 3-D test structure used in this study is a symmetric staircase structure consisting
of nine steps where the width and length of each step are 1.5 ?m and 50 ?m, respectively.
The substrate system is composed of 1000 nm poly (methyl methacrylate) (PMMA) on Si
and the beam energy is assumed to be 50 KeV .
18
In Figures 3.4(a) and (b), the resist proflles obtained without and with width adjustment
are provided along with the step-edge (?Wi) and width (?Wi) deviations in the flgure caption.
Note that in order to show the detail, the vertical (depth) dimension is scaled up (relative
to the horizontal dimension). It can be seen that the resist proflle before width adjustment
shows a signiflcant edge deviation of each step and a larger deviation for a deeper step. The
reason why the center step has the largest width error (deviation) is that the directions of
lateral development on the left and right edges of the center step are opposite. Therefore, the
step-width deviation (?Wi) for the center step is twice the step-edge deviation (?Wi) in this
symmetric staircase. However, for other steps, the step-edge deviations at the left and right
edges are in the same direction. Hence, the step-width deviation of a step is smaller than
the step-edge deviation of either edge. Also, the higher exposure at the center step helps to
make ?Wi larger than those for other steps. The flnal proflle after step-width adjustment,
combined with the dose control scheme in Figure 3.4(b), is signiflcantly closer to the target
proflle and shows a substantial improvement in step-width accuracy.
In Figure 3.5, the simulation results for a symmetric staircase structure with a smaller
step-width of 1 ?m and a thinner resist of 500 nm are provided. It is observed that the
step-edge and width deviations are smaller than those for the thicker resist of 1000 nm since
the developing time is shorter, i.e., less time for lateral development. Again, through step-
width adjustment, a resist proflle much closer to the target proflle has been obtained. Note
that the cell removal model is carried out to simulate the remaining resist proflle of staircase
structures only and also to verify the accuracy of the step-width adjustment scheme, and it
is not used for the step-width deviation estimation.
3.4.2 Experimental Results
The symmetric nine-step staircase structure with the step-width of 1.5 ?m adopted in
the simulation study has been fabricated with and without step-width adjustment. The
substrate system was prepared by spin-coating a Si wafer with 1000 nm PMMA and soft-
baked at 160 oC for 1 minute. The structure was written using an Elionix ELS-7000 e-beam
19
tool with acceleration voltage of 50 KeV and beam current of 100 pA. The sample was
developed in MIBK:IPA = 1:1 for 30 seconds. The remaining resist was coated with 10 nm
Pt before the cross section was imaged by a FEI FE-SEM (Sirion). For easier inspection of
the cross section, the length of the structure was increased to 500 ?m. The SEM images of
the cross section are provided in Figure 3.6.
It is seen in Figure 3.6(a) that the width of the center step is 11.3% wider than the
target width. As mentioned in Section 2.6, the width deviation for the center step was
estimated to be 16% in the simulation, which is reasonably close to the experimental result.
As shown in Figure 3.6(b), the step-width adjustment not only reduced the width deviation
greatly, but also decreased the step-height deviation, ?di, making the step-height closer to
the target height of 200 nm. In Figure 3.7, the step-width deviations measured in the
simulation and experimental results, before and after step-width adjustment, are plotted. In
the experimental result, the maximum step-width deviation was reduced from 170 nm down
to 10 nm and the average step-width deviation was from 37.8 nm to 2.2 nm where the target
width is 1.5 ?m. Also, the simulation result closely agrees with the experimental result,
which well demonstrates the accuracy of our simulation model and step-width adjustment
scheme.
20
X (?m)Resist depth (nm) 1.5 3 4.5 6 7.5 9 10.5 12 13.5
200
400
600
800
1,000
(a)
X (?m)Resist depth (nm) 1.5 3 4.5 6 7.5 9 10.5 12 13.5
200
400
600
800
1,000
(b)
Figure 3.4: (a) Step-edge deviation: f?Wig = f0, 45, 70, 90, 125, 125, 90, 70, 45, 0 nmg
Step-width deviation: f?Wig = f45, 25, 20, 35, 250, 35, 20, 25, 45 nmg (b) Step-edge
deviation: f?Wig = f0, 0, 0, 0, 0, 0, 0, 0, 0, 0 nmg Step-width deviation: f?Wig =
f0, 0, 0, 0, 0, 0, 0, 0, 0 nmg Remaining resist proflles obtained (through simulation) for
the staircase structure (a) before adjustment and (b) after step-width and dose adjustments
where target step-width: 1.5 ?m, target step-height: 200 nm, resist thickness: 1000 nm on
Si (50 KeV). A deviation less than 1 nm is rounded to zero.
21
X (?m)Resist depth (nm) 1 2 3 4 5 6 7 8 9
100
200
300
400
500
(a)
X (?m)Resist depth (nm) 1 2 3 4 5 6 7 8 9
100
200
300
400
500
(b)
Figure 3.5: (a) Step-edge deviation: f?Wig = f0, 15, 25, 35, 60, 60, 35, 25, 15, 0 nmg
Step-width deviation: f?Wig = f15, 10, 10, 25, 120, 25, 10, 10, 15 nmg (b) Step-edge
deviation: f?Wig = f0, 0, 0, 0, 0, 0, 0, 0, 0, 0 nmg Step-width deviation: f?Wig =
f0, 0, 0, 0, 0, 0, 0, 0, 0 nmg Remaining resist proflles obtained (through simulation) for
the staircase structure (a) before adjustment and (b) after step-width and dose adjustments
where target step-width: 1.0 ?m, target step-height: 100 nm, resist thickness: 500 nm on
Si (50 KeV). A deviation less than 1 nm is rounded to zero.
22
(a)
(b)
Figure 3.6: Experimental results (a) before step-width adjustment and (b) after step-width
adjustment where target step-width: 1.5 ?m, target step-height: 200 nm, resist thickness:
1000 nm PMMA on Si (50 KeV).
23
1 2 3 4 50
50
100
150
200
250
Step
Step?width deviation (nm)  
 
Simulation ?wi
 Experimental ?wi
(a)
1 2 3 4 50
2
4
6
8
10
Step
Step?width deviation (nm)  
 
Simulation ?wi
 Experimental ?wi
(b)
Figure 3.7: Step-width deviations in simulation and experimental results (a) before step-
width adjustment and (b) after step-width adjustment for 1000 nm PMMA (50 KeV).
24
Chapter 4
Controlling Sidewall of Resist Proflle
A certain type of resist proflle is desired depending on the subsequent process following
resist development [6]. For example, an undercut proflle is required for lift-ofi and a straight
vertical sidewall for etching. As the feature size is reduced down to nanoscale, the aspect
ratio of developed feature in the resist proflle becomes larger even for a thin resist. This
makes a small variation in the sidewall slope cause a relatively large critical-dimension (CD)
error. Therefore, it is important to have a su?cient control over the sidewall shape in the
resist proflle. The sidewall shape obtained through e-beam lithographic process depends
on factors such as exposure (energy deposited in resist) distribution, developing time, etc.
Varying developing time is a passive approach in that the spatial exposure distribution is
set, and therefore has a limited controllability. Controlling the exposure distribution, more
precisely the 3-D distribution of exposure, enables a more explicit method to achieve a target
sidewall. Nevertheless, in most of the previous work, only the dose level was varied with a
uniform dose within a feature, to achieve difierent shapes of sidewall, and the 3-D exposure
distribution was not considered. Changing the level of uniform dose only scales the exposure
distribution without altering the spatial distribution and therefore does not fully utilize the
available controllability of exposure distribution [42]. Here, a general-purpose optimization
method, Simulated Annealing (SA), is adopted in determining the dose distribution required
for target sidewall shapes.
4.1 3-D Model
The sidewall of resist proflle for a line pattern is considered as illustrated in Figure
2.3(a) and its cross section plane (X-Z plane) is shown in Figure 2.3(b).
25
4.1.1 Developing Rate
The developer and developing time used to fabricate the single line are MIBK:IBA = 1:2
and 40 sec, respectively, which are difierent from those in fabrication of staircase structures,
thustheexposure-to-rate conversionformulaneedstobere-derived. Atthecenterofaline(in
the cross section plane) where the exposure is highest when a uniform dose is given to the line,
resist development progresses mainly downward (i.e., along the vertical dimension) such that
the lateral development may be ignored. Exploiting this property, the conversion formula is
derived using a part of the 3rd-order polynomial curve. It models the cross section of resist
layer by a 2-D array of blocks within each of which the exposure and therefore developing
rate are assumed to be constant. Using the remaining resist proflle from experiments, the
conversion formula is calibrated iteratively by modeling the developing rate block by block
of the vertical column at the center of the line.
The following conversion formula was obtained:
r = 2:8?10?29 ?e3 +4:9?10?19 ?e2 +0:39?10?8 ?e (4.1)
where e is in eV/?m3 and r is in nm/min.
4.1.2 Simulation of Resist Development
For the sidewall shape control, the fast and accurate resist development simulation
mentioned in Section 2.6.2 is employed to flnd the optimal dose distribution to achieve the
target sidewall shape because of its high e?ciency. When a small region has a much higher
exposure than its surrounding regions, its efiective developing rate is signiflcantly lower than
the nominal rate (given by Eqn. 4.1 due to the aspect-ratio-dependent development). To
re ect this efiect in development simulation, the developing rate is adjusted according to the
spatial distribution of exposure before the simulation.
4.2 Sidewall Control
Given a developer and a developing time, the resist proflle depends on the exposure
distribution e(x;z). Therefore, one may attempt to control e(x;z) in order to achieve a target
26
resist proflle. When a substrate system is given, e(x;z) is determined by the distribution of
e-beam dose within the feature, i.e., a line. The e-beam dose is varied (controlled) only along
the width dimension, i.e., X-axis and therefore the dose distribution is denoted by d(x). The
feature considered in this study is a long line and the cross section of resist proflle at the
center of the line is characterized by the line widths in the top, middle and bottom layers
as illustrated in Figure 4.1. Let rxi and pxi represent the target and actual widths in the
ith layer, respectively. Then, the optimization problem for sidewall control can be deflned
as flnding d(x) such that the cost function maxi(jrxi ?pxij) is minimized.
Figure 4.1: An illustration of sidewall shape speciflcation in the cross section: rxi and pxi
are the target and actual widths of line in the ith layer of resist, respectively. The cost
function is deflned as C = maxi(jrxi ?pxij).
In order to avoid an impractically long computation time, the line is partitioned into n
regions along the length dimension as shown in Figure 4.2(b) and a dose for each region is
to be determined. That is, the solution from the optimization is a dose set (d1;d2;???;dn)
where dj is the dose for the jth region. A fundamental di?culty of this optimization is that
the optimal dose for a region has con icts among layers, i.e., the dose required for a layer
may be difierent from that for another layer. Also, the optimal dose for a region depends
on the doses of the other regions. In this study, the general-purpose optimization method of
27
Simulated Annealing (SA) is adopted, which perturbs the doses of multiple regions in each
iteration to flnd a globally optimal solution [49].
Figure 4.2: Dose distribution and the corresponding sidewall shape: (a) a uniform dose
distribution and (b) the spatially-controlled dose distribution.
A single dose for all regions, minimizing the cost function, is flrst determined as an initial
solution. For evaluation of the cost function, the exposure distribution in the cross section
is computed through the convolution between the dose distribution and the point spread
function. Then, the resist development simulation is carried out to measure the dimensional
errors in terms of line widths, i.e., jrxi?pxij. The main optimization procedure of SA starts
from the initial solution and iteratively derives the optimal or an acceptable solution. The
 owchart of SA is given in Figure 4.3 and the steps in SA are described below. The solution
obtained in the kth iteration is denoted by S0(k) = (d(k)1 ;d(k)2 ;:::;d(k)i ;:::;d(k)n ) where d(k)i is the
dose for the ith region, derived in the kth iteration.
28
Figure 4.3: The  owchart of SA (simulated annealing) process
Step 1: Initially, the solution S is set to S0(0)=(d(0)1 ;d(0)2 ;:::;d(0)i ;:::;d(0)n ), and the temperature
T to a high value of T0. A possible initial dose distribution is a uniform distribution,
i.e., d(0)i = d(0)j for all i;j. The cost function C = maxi(jrxi ?pxij) is evaluated based
on S0(0) through resist development simulation. Let C0(0) denote the value of the cost
function for S0(0).
Step 2: Randomly perturb the current solution (spatial dose distribution) S(k) to a potential
new solution S0(k+1) = S(k)+(?d(k+1)1 ;:::;?d(k+1)i ;:::;?d(k+1)n ), where S(k) is the ac-
cepted dose distribution in the kth iteration and ?d(k+1)i is the amount of dose change
for the ith region in the (k + 1)th iteration. Note that the doses of all regions are
adjusted as illustrated in Figure 4.4. In the case of single line, the dose distribution
29
must be symmetric with respect to the center of line. Therefore, only n+12 ?d(k+1)i ?s
need to be determined. Determination of ?d(k+1)i may be guided by a certain heuristic.
In this study, ?d(k+1)i is computed as follows
?d(k+1)i =
 
0:5?(dmax ?dmin ?dminJump)?(1+cos(j ??J ))+dminJump
?
?(r?0:5)
(4.2)
where dmax and dmin are the upper and lower limits of dose allowed, dminJump is the
minimum dose step of ?d(k+1)i , j is the index for the jth temperature decrement from
T0 to the current T, J is the total number of temperature decrements and r is a random
number ranging [0;1].
Figure 4.4: During the SA process, the doses of all regions of a line feature are adjusted. The
solid line and dashed lines represent the dose distribution before and after dose adjustment.
Note that the dose step range is adjusted (decreased) as the temperature is decreased
as shown in Figure 4.5. The cost function C is evaluated for S0(k+1) to obtain its cost
C0(k+1).
Step 3: When ?C = C0(k+1)?C(k) < 0 where C(k) is the value of cost function for S(k), S0(k+1)
is accepted to become S(k+1). If ?C > 0, S0(k+1) is still accepted with the probability
30
10?410?2100101 10?1 10?30
200
400
600
800
1000
1200
Temperature (T)
Dose step range (
?C/cm
2 )
Figure 4.5: Dose step range vs temperature in the SA process.
of exp(??CT ). This acceptance of a worse solution enables the hill-climbing capability
of SA toward the globally optimal solution. Otherwise, S0(k+1) is rejected in which
case S(k) becomes S(k+1). If the number of successive rejections Nrej exceeds a certain
threshold, go to Step 4. Otherwise, go to Step 2.
Step 4: The temperature T is lowered according to T ? fi ? T where 0 < fi < 1. That
is, as the SA progresses, a worse solution is accepted less (since it is likely that the
current solution is closer to the optimal solution). Go to Step 2 if T is above the flnal
temperature. Otherwise, go to Step 5.
Step 5: The current solution is taken as the flnal solution (dose distribution).
Constraints
The optimization of the dose distribution may be done with or without constraints such
as total dose, developing time, etc. It is always desirable to minimize the time to expose
a pattern from the viewpoint of throughput. The exposing time is mainly proportional to
the total dose to be given to the pattern. Also, the smaller the total dose is, the lower the
charging efiect is. Hence, a dose distribution of which the total dose is smaller is better
as long as it achieves an equivalent quality of the resist proflle. In most of our study, the
constraint of the same total (average) dose was imposed, i.e., DW = RW0 d(x)dx where D is
a certain dose level and W is the line width as shown in Figure 4.2. In other words, the same
31
total dose is redistributed over the line (feature) through optimization in order to achieve a
certain target proflle.
4.3 Results and Discussion
Three difierent dose distributions were considered in computer simulation as shown in
Figure 4.6. One, referred to as Distribution-A, is a uniform distribution. Another, referred
to as Distribution-B, is the one where the edge dose is moderately larger than the center
dose. The other, referred to as Distribution-C, has the edge dose much larger than the center
dose.
100 120 140 160 180 2000
200
400
600
800
X(nm)
Dose (
?C/cm
2 )
 
 Distribution?A
Distribution?BDistribution?C
Figure 4.6: Three dose distributions of Distribution-A, Distribution-B and Distribution-C
4.3.1 Simulation Results
The test feature used in this study is a single line where the width and length of the line
are 100 nm and 50 ?m, respectively. The substrate system is composed of 300 nm PMMA
on Si and the beam energy is assumed to be 50 KeV. The total (or average) dose is flxed in
each set of results unless specifled otherwise.
32
The relationship among the dose distribution, total dose, and developing time in terms
of their efiects on the sidewall shape has been analyzed through simulation. The sidewall
shapes obtained for the three difierent dose distributions (Figure 4.6) with the total dose
and developing time flxed are shown in Figure 4.7 and the line-width measurements are
provided in Table 4.1. It can be seen that a difierent dose distribution leads to a difierent
sidewall shape. For a vertical sidewall, the Distribution-B minimizes the width error (among
the three). The developing time required for a target sidewall shape (rx1 = 130nm, rx5 =
130nm, rx10 = 130nm) was derived for each of the dose distributions in Figure 4.6. The
Distribution-B requires the shortest developing time to achieve the sidewall shape closest to
the target one (refer to Table 4.2). The three dose distributions were scaled by a certain
factor to achieve the target sidewall shape and the results are provided in Table 4.3. The
Distribution-B requires the lowest total (average) dose while minimizing the width error.
X (nm)
Resist Depth (
nm)
 
 
0 50 100150200250300350400450500
50
100
150
200
250
300
Distribution?ADistribution?B
Distribution?C
Figure 4.7: The remaining resist proflles (sidewall shapes) of Distribution-A, Distribution-B
and Distribution-C.
In Figure 4.8, the remaining resist proflles obtained from the three difierent types of
spatial dose distributions are provided for the average dose of 500 ?C=cm2. The target
sidewall shape is vertical. When the dose is not controlled, i.e., a constant dose of 500
33
Resist Proflle (nm)
Dose Distribution Average Dose Developing Time Line Width
(?C=cm2) (sec) px1 px5 px10
A:Dashed curve 500.0 40.0 124.4 117.0 71.4
B:Solid curve 500.0 40.0 129.8 126.0 115.0
C:Dotted curve 500.0 40.0 131.4 124.0 0.0
Table 4.1: Efiects of the dose distribution on the sidewall shape with the total (average)
dose and developing time flxed.
Resist Proflle (nm)
Dose Distribution Average Dose Developing Time Line Width
(?C=cm2) (sec) px1 px5 px10
A:Dashed curve 500.0 66.0 131.7 131.0 129.0
B:Solid curve 500.0 51.0 133.0 132.0 129.0
C:Dotted curve 500.0 60.0 137.5 135.0 128.0
Table 4.2: The developing time required to achieve the same (equivalent) sidewall shape
with the total (average) dose flxed.
?C=cm2 (Distribution-A), the sidewall shape obtained is of overcut as can be seen in Figure
4.8(a), which is signiflcantly difierent from the target one. The sidewall shape obtained
with spatial dose control (Distribution-B), shown in Figure 4.8(b), is much closer to the
target sidewall shape. With a constant dose, developing rates in edge regions of the line
are smaller than those at the center region, so the resist in edge regions is developed slower
vertically, leading to an overcut. When the dose is higher in edge regions of the line than
in the center region as in the Distribution-B, the developing rate is higher in edge regions
which causes lateral development at lower layers to start earlier. And also the exposure in
Resist Proflle (nm)
Dose Distribution Average Dose Developing Time Line Width
(?C=cm2) (sec) px1 px5 px10
A:Dashed curve 640.0 40.0 130.0 130.0 129.0
B:Solid curve 560.0 40.0 131.6 131.0 129.0
C:Dotted curve 610.0 40.0 134.8 134.0 128.0
Table 4.3: The total (average) dose required to achieve the same (equivalent) sidewall shape
with the developing time flxed.
34
unexposed regions tends to increase with depth. Therefore, lateral development following
vertical development in the edge region catches up with vertical development right outside
the edge region, leading to a more vertical sidewall shape. However, if the edge dose is
increased beyond a certain level (with the average dose flxed) as in the Distribution-C, the
efiective developing rate is decreased signiflcantly. This is due to the fact that the edge
developing rate is much higher than that in its surrounding regime. Hence, the sidewall
shape becomes overcut as seen in Figure 4.8(c).
The same set of the results for the average dose of 525 ?C=cm2 are provided in Figure
4.9. The sidewall for the increased constant dose, in Figure 4.9(a), is more vertical than
that in Figure 4.8(a), but still not so vertical as that in Figure 4.8(b). As in the case of the
average dose of 525 ?C=cm2, the spatial dose control with a small dose difierence between
the edge and center regions results in a much more vertical sidewall as can be seen in Figure
4.9(b), which is almost the same as that in Figure 4.8(b). Again, too large a dose difierence
between the edge and center regions leads to an overcut shown in Figure 4.9(c).
35
X (nm)
Resist Depth (
nm)
0 50100150200250300350400450500
50100
150200
250300
(a)
X (nm)
Resist Depth (
nm)
0 50100150200250300350400450500
50100
150200
250300
(b)
X (nm)
Resist Depth (
nm)
0 50100150200250300350400450500
50100
150200
250300
(c)
Figure 4.8: Simulation results with the same average dose 500 ?C=cm2 (a) simulation
result for Distribution-A (b) simulation result for Distribution-B (c) simulation result for
Distribution-C where developing time: 40 sec, MIBK:IPA=1:2, 300 nm PMMA on Si (50
KeV).
36
X (nm)
Resist Depth (
nm)
0 50100150200250300350400450500
50100
150200
250300
(a)
X (nm)
Resist Depth (
nm)
0 50100150200250300350400450500
50100
150200
250300
(b)
X (nm)
Resist Depth (
nm)
0 50100150200250300350400450500
50100
150200
250300
(c)
Figure 4.9: Simulation results with the same average dose 525 ?C=cm2 (a) simulation
result for Distribution-A (b) simulation result for Distribution-B (c) simulation result for
Distribution-C where developing time: 40 sec, MIBK:IPA=1:2, 300 nm PMMA on Si (50
KeV).
37
4.4 Trend Analysis of Sidewall Control
In this section, the relationship among dose distribution, developing time, average dose,
resist thickness and the sidewall shape is analyzed through computer simulation. A line is
partitioned into 5 regions along the length dimension and the dose distribution over the 5
regions is symmetric. Therefore, in the result tables, only the doses of 3 regions are provided
(d1, d2, and d3 are the doses of the edge, middle and center regions, respectively).
4.4.1 The Efiect of Developing Time on Dose Distribution
For the same average dose and thickness, the developing time afiects the dose distribu-
tion within the line, required to achieve the same sidewall shape of resist proflle. As shown
in Table 4.4, for 300 nm PMMA on Si , to achieve the equivalent sidewall shape with the
average dose flxed, the dose difierence between edge and center regions becomes smaller for
a longer developing time. In general, the edge dose d1 decreases while the center dose d3
tends to increase with the increase of developing time. In addition, difierent combinations
(small average dose/longer developing time or shorter developing time/larger average dose)
can give the same resist proflle.
Dose Distribution Target Proflle (nm) Actual Proflle (nm)
Average Dose Developing Time Line Width Line Width
(?C=cm2) (sec) d1 d2 d3 rx1 rx5 rx10 px1 px5 px10
42.0 702.49 575.19 144.65 130.0 130.0 130.0 130.7 130.0 127.0
540.0 44.0 701.85 518.10 260.09 130.0 130.0 130.0 131.6 131.0 129.0
46.0 676.45 486.47 374.16 130.0 130.0 130.0 131.6 131.0 129.0
40.0 748.64 583.04 136.64 130.0 130.0 130.0 131.6 131.0 129.0
560.0 42.0 687.84 631.19 161.94 130.0 130.0 130.0 130.7 131.0 130.0
44.0 668.40 573.80 315.60 130.0 130.0 130.0 130.7 131.0 129.0
38.0 752.28 594.93 205.58 130.0 130.0 130.0 130.7 131.0 129.0
580.0 40.0 691.11 673.72 170.34 130.0 130.0 130.0 130.7 130.0 130.0
42.0 666.78 648.40 269.65 130.0 130.0 130.0 130.7 130.0 130.0
36.0 753.61 654.57 183.65 130.0 130.0 130.0 130.7 130.0 130.0
600.0 38.0 717.63 651.94 260.86 130.0 130.0 130.0 130.7 130.0 130.0
40.0 688.03 637.71 348.52 130.0 130.0 130.0 130.7 130.0 127.0
Table 4.4: Comparison of dose distributions with the same target resist proflle (vertical
sidewall) for line width of 100 nm and the same average dose (resist thickness: 300 nm).
38
For the resist thicknesses of 100 nm and 500 nm, the above trends still hold, as shown
in Tables 4.5 and 4.6. For a thinner resist, it is easier to get vertical sidewalls and, from
Table 4.5, it is seen that the slightly undercut sidewall can also be achieved. While for a
thicker resist, it is imperative to increase the average dose signiflcantly to fully develop it;
the actual resist proflle is far from the target proflle. That is, it is more di?cult to control
the sidewall shape of a thicker resist. Note that the exposure varies more along the depth
dimension when the resist is thicker.
Dose Distribution Target Proflle (nm) Actual Proflle (nm)
Average Dose Developing Time Line Width Line Width
(?C=cm2) (sec) d1 d2 d3 rx1 rx5 rx10 px1 px5 px10
42.0 506.21 376.48 234.62 120.0 120.0 120.0 115.9 117.0 117.0
400.0 44.0 485.31 373.88 281.63 120.0 120.0 120.0 115.9 117.0 117.0
46.0 451.63 377.07 342.61 120.0 120.0 120.0 115.9 117.0 117.0
40.0 552.57 374.78 245.31 120.0 120.0 120.0 116.6 117.0 117.0
420.0 42.0 496.28 439.09 229.25 120.0 120.0 120.0 115.9 117.0 117.0
44.0 474.12 394.71 362.33 120.0 120.0 120.0 115.9 117.0 117.0
38.0 585.87 423.63 181.01 120.0 120.0 120.0 116.8 117.0 118.0
440.0 40.0 516.53 437.32 292.30 120.0 120.0 120.0 115.9 117.0 118.0
42.0 487.20 448.86 327.88 120.0 120.0 120.0 115.9 117.0 117.0
Table 4.5: Comparison of dose distributions with the same target resist proflle (vertical
sidewall) for line width of 100 nm and the same average dose (resist thickness: 100 nm).
Dose Distribution Target Proflle (nm) Actual Proflle (nm)
Average Dose Developing Time Line Width Line Width
(?C=cm2) (sec) d1 d2 d3 rx1 rx5 rx10 px1 px5 px10
42.0 786.41 721.83 133.53 130.0 130.0 130.0 124.2 131.0 129.0
630.0 44.0 737.85 734.31 205.67 130.0 130.0 130.0 124.2 131.0 130.0
46.0 712.81 711.03 302.33 130.0 130.0 130.0 125.2 131.0 129.0
40.0 807.06 723.35 189.17 130.0 130.0 130.0 124.2 131.0 130.0
650.0 42.0 775.89 666.32 365.57 130.0 130.0 130.0 124.2 131.0 129.0
44.0 733.35 710.53 362.24 130.0 130.0 130.0 125.2 131.0 130.0
38.0 827.34 734.13 227.06 130.0 130.0 130.0 124.2 131.0 131.0
670.0 40.0 779.76 733.90 322.67 130.0 130.0 130.0 124.2 131.0 131.0
42.0 761.60 709.12 408.56 130.0 130.0 130.0 124.2 131.0 131.0
Table 4.6: Comparison of dose distributions with the same target resist proflle (vertical
sidewall) for line width of 100 nm and the same average dose (resist thickness: 500 nm).
39
From Tables 4.4-4.6, the efiect of developing time on dose distribution is analyzed for
the vertical sidewall shape. Given a target undercut sidewall shape, as shown in Table 4.7,
a longer developing time results in a smaller dose difierence between edge and center regions
with the average dose and resist thickness (300 nm) flxed. Compared with the developing
time required for a vertical sidewall in Table 4.4, a relatively longer developing time is desired
in order to achieve an undercut sidewall for the same resist thickness.
Dose Distribution Target Proflle (nm) Actual Proflle (nm)
Average Dose Developing Time Line Width Line Width
(?C=cm2) (sec) d1 d2 d3 rx1 rx5 rx10 px1 px5 px10
50.0 750.56 617.48 163.92 130.0 135.0 140.0 134.0 136.0 139.0
580.0 52.0 740.07 545.74 328.40 130.0 135.0 140.0 134.0 136.0 139.0
54.0 717.00 492.97 480.07 130.0 135.0 140.0 134.0 136.0 139.0
48.0 744.14 675.88 159.96 130.0 135.0 140.0 134.0 136.0 139.0
600.0 50.0 736.32 624.25 278.85 130.0 135.0 140.0 134.0 136.0 139.0
52.0 725.49 530.58 487.84 130.0 135.0 140.0 134.0 136.0 139.0
46.0 788.23 678.80 165.94 130.0 135.0 140.0 134.0 136.0 140.0
620.0 48.0 735.57 703.73 221.40 130.0 135.0 140.0 134.0 136.0 140.0
50.0 730.94 623.08 391.97 130.0 135.0 140.0 134.0 136.0 140.0
Table 4.7: Comparison of dose distributions with the same target resist proflle (undercut
sidewall) for line width of 100 nm and the same average dose (resist thickness: 300 nm).
40
4.4.2 The Efiect of Average Dose on Dose Distribution
The average dose afiects the spatial distribution of doses with the target proflle and
developing time flxed. The dose difierence between edge and center regions becomes smaller
for a larger average dose, which can be seen from Tables 4.8-4.11. That is, the dose distri-
bution becomes  atter as the average dose increases. Therefore, one of the ways to achieve
a vertical sidewall or undercut sidewall is to increase the dose level of uniform distribution,
which however requires a longer exposing time and leads to a larger charging efiect.
Dose Distribution Target Proflle (nm) Actual Proflle (nm)
Developing Time Average Dose Line Width Line Width
(sec) (?C=cm2) d1 d2 d3 rx1 rx5 rx10 px1 px5 px10
580.0 752.28 594.93 205.58 130.0 130.0 130.0 130.7 131.0 129.0
38.0 600.0 717.63 651.94 260.86 130.0 130.0 130.0 130.7 130.0 130.0
620.0 699.56 678.02 344.83 130.0 130.0 130.0 130.7 130.0 130.0
560.0 748.64 583.04 136.64 130.0 130.0 130.0 131.6 131.0 129.0
40.0 580.0 691.11 673.72 170.34 130.0 130.0 130.0 130.7 130.0 130.0
600.0 688.03 637.71 348.52 130.0 130.0 130.0 130.7 130.0 127.0
540.0 702.49 575.19 144.65 130.0 130.0 130.0 130.7 130.0 127.0
42.0 560.0 687.84 631.19 161.94 130.0 130.0 130.0 130.7 131.0 130.0
580.0 666.78 648.40 269.65 130.0 130.0 130.0 130.7 130.0 130.0
Table 4.8: Comparison of dose distributions with the same target resist proflle (vertical
sidewall) for line width of 100 nm and the same developing time (resist thickness: 300 nm).
Dose Distribution Target Proflle (nm) Actual Proflle (nm)
Developing Time Average Dose Line Width Line Width
(sec) (?C=cm2) d1 d2 d3 rx1 rx5 rx10 px1 px5 px10
420.0 552.57 374.78 245.31 120.0 120.0 120.0 116.6 117.0 117.0
40.0 440.0 516.53 437.32 292.30 120.0 120.0 120.0 115.9 117.0 118.0
460.0 493.25 474.07 365.37 120.0 120.0 120.0 115.9 117.0 117.0
400.0 506.21 376.48 234.62 120.0 120.0 120.0 115.9 117.0 117.0
42.0 420.0 496.28 439.09 229.25 120.0 120.0 120.0 115.9 117.0 117.0
440.0 487.20 448.86 327.88 120.0 120.0 120.0 115.9 117.0 117.0
380.0 491.50 323.89 269.23 120.0 120.0 120.0 115.9 116.0 116.0
44.0 400.0 485.31 373.88 281.63 120.0 120.0 120.0 115.9 117.0 117.0
420.0 474.12 394.71 362.33 120.0 120.0 120.0 115.9 117.0 117.0
Table 4.9: Comparison of dose distributions with the same target resist proflle (vertical
sidewall) for line width of 100 nm and the same developing time (resist thickness: 100 nm).
41
Dose Distribution Target Proflle (nm) Actual Proflle (nm)
Developing Time Average Dose Line Width Line Width
(sec) (?C=cm2) d1 d2 d3 rx1 rx5 rx10 px1 px5 px10
650.0 807.06 723.35 189.17 130.0 130.0 130.0 124.2 131.0 130.0
40.0 670.0 779.76 733.90 322.67 130.0 130.0 130.0 124.2 131.0 131.0
690.0 775.97 673.47 551.12 130.0 130.0 130.0 124.2 131.0 129.0
630.0 786.41 721.83 133.53 130.0 130.0 130.0 124.2 131.0 129.0
42.0 650.0 775.89 666.32 365.57 130.0 130.0 130.0 124.2 131.0 129.0
670.0 761.60 709.12 408.56 130.0 130.0 130.0 124.2 131.0 131.0
610.0 745.71 726.24 106.09 130.0 130.0 130.0 124.2 131.0 128.0
44.0 630.0 737.85 734.31 205.67 130.0 130.0 130.0 124.2 131.0 130.0
650.0 733.35 710.53 362.24 130.0 130.0 130.0 125.2 131.0 130.0
Table 4.10: Comparison of dose distributions with the same target resist proflle (vertical
sidewall) for line width of 100 nm and the same developing time (resist thickness: 500 nm).
4.4.3 The Efiect of Developing Time on Sidewall Shape
The developing time has a signiflcant efiect on the sidewall shape of a line. In general,
a longer developing time results in a more vertical or undercut sidewall for a given dose
distribution, as shown in Tables 4.12 and 4.13. The dose distributions in Tables 4.12 and
4.13 are derived from the SA method with the target proflle: rx1 = 130nm, rx5 = 130nm,
rx10 = 130nm and the developing time: 40.0 sec. Based on these dose distributions, the
developing time is varied to see its efiect on the sidewall shape. A longer developing time
allows a longer time for lateral development. And the developing rate in the unexposed region
increases with depth, therefore, the lateral development following vertical development in
the unexposed region catches up with vertical development right outside the edge region,
resulting in a more vertical sidewall shape and eventually an undercut.
4.4.4 The Efiect of Dose Distribution on Sidewall Shape
In Table 4.14, the dose distributions which minimize the deviation from the target
sidewall shape and were obtained by the SA method are listed. For each thickness of resist,
the combination of the developing time and minimum average dose is varied. In order to
quantify the characteristic of dose distribution, d1, d2 and d3 normalized by d1 are also
included in the table. It is observed that d2 becomes closer to d1 with d1 still greater than d2
42
Dose Distribution Target Proflle (nm) Actual Proflle (nm)
Developing Time Average Dose Line Width Line Width
(sec) (?C=cm2) d1 d2 d3 rx1 rx5 rx10 px1 px5 px10
600.0 744.14 675.88 159.96 130.0 135.0 140.0 134.0 136.0 139.0
48.0 620.0 735.57 703.73 221.40 130.0 135.0 140.0 134.0 136.0 140.0
640.0 728.29 665.42 412.58 130.0 135.0 140.0 134.0 136.0 140.0
580.0 750.56 617.48 163.92 130.0 135.0 140.0 134.0 136.0 139.0
50.0 600.0 736.32 624.25 278.85 130.0 135.0 140.0 134.0 136.0 139.0
620.0 730.94 623.08 391.97 130.0 135.0 140.0 134.0 136.0 140.0
560.0 780.71 530.60 177.39 130.0 135.0 140.0 135.0 136.0 138.0
52.0 580.0 740.07 545.74 328.40 130.0 135.0 140.0 134.0 136.0 139.0
600.0 725.49 530.58 487.84 130.0 135.0 140.0 134.0 136.0 139.0
Table 4.11: Comparison of dose distributions with the same target resist proflle (undercut
sidewall) for line width of 100 nm and the same developing time (resist thickness: 300 nm).
Dose Distribution Actual Proflle (nm)
Developing Time Line Width
(sec) d1 d2 d3 px1 px5 px10
40.0 748.64 583.04 136.64 131.6 131.0 129.0
46.0 748.64 583.04 136.64 132.6 133.0 133.0
48.0 748.64 583.04 136.64 133.8 134.0 135.0
60.0 748.64 583.04 136.64 136.0 139.0 143.0
Table 4.12: The efiect of developing time on sidewall shape with the average dose (560.0
?C=cm2) and dose distribution flxed (resist thickness: 300 nm).
as the resist thickness increases. Also, d3 is relatively smaller for a thicker resist. In addition,
it can be seen that the normalized d1, d2 and d3 remain similar within each thickness of resist.
43
Dose Distribution Actual Proflle (nm)
Developing Time Line Width
(sec) d1 d2 d3 px1 px5 px10
40.0 807.06 723.35 189.17 124.2 131.0 130.0
42.0 807.06 723.35 189.17 124.2 132.0 133.0
48.0 807.06 723.35 189.17 126.0 135.0 141.0
60.0 807.06 723.35 189.17 129.6 141.0 152.0
Table 4.13: The efiect of developing time on sidewall shape with the average dose (650.0
?C=cm2) and dose distribution flxed (resist thickness: 500 nm).
Resist Average Developing Dose Distribution Target Proflle (nm) Actual Proflle (nm)
Thickness dose time Ratio Line Width Line Width
(nm) (?C=cm2) (sec) d1 d2 d3 d1d1 : d2d1 : d3d1 rx1 rx5 rx10 px1 px5 px10
400.0 42.0 506.2 376.5 234.6 1:0.744:0.464 120.0 120.0 120.0 115.9 117.0 117.0
100 420.0 40.0 552.6 374.8 245.3 1:0.678:0.444 120.0 120.0 120.0 116.6 117.0 117.0
440.0 38.0 585.9 423.6 181.0 1:0.723:0.309 120.0 120.0 120.0 116.8 117.0 118.0
540.0 42.0 702.5 575.2 144.7 1:0.819:0.206 130.0 130.0 130.0 130.7 130.0 127.0
300 560.0 40.0 748.6 583.0 136.6 1:0.779:0.183 130.0 130.0 130.0 131.6 131.0 129.0
580.0 38.0 752.3 594.9 205.6 1:0.791:0.273 130.0 130.0 130.0 130.7 131.0 129.0
630.0 42.0 786.4 721.8 133.5 1:0.918:0.170 130.0 130.0 130.0 124.2 131.0 129.0
500 650.0 40.0 807.1 723.4 189.2 1:0.896:0.234 130.0 130.0 130.0 124.2 131.0 130.0
670.0 38.0 827.3 734.1 227.1 1:0.887:0.274 130.0 130.0 130.0 124.2 131.0 131.0
Table 4.14: Comparison of dose distributions required to achieve a target sidewall shape for
difierent resist thicknesses. For each given developing time, the average dose is minimized.
44
Chapter 5
Concluding Remarks and Future Study
In this thesis, the related problems of step-width adjustment and sidewall control in
electron-beam lithography have been studied, based on the common framework of 3-D ex-
posure model and resist development simulation.
Due to lateral development of step sidewalls in a staircase structure during the resist
development process, the step-widths in the flnal resist proflle may be substantially difierent
from the target widths. A practical method for adjusting step-widths, developed for stair-
case structures, is described. The method is based on a 3-D exposure model and utilizes
the exposure-to-rate conversion formula derived from experimental results. The method es-
timates step-width deviation and then compensates for the deviation by a combination of
width adjustment and dose control, which achieves a flne control of step-width and height.
Also, the issue of sidewall control has been further investigated based on the results from
the previous work. A line is partitioned into regions along the length dimension and a dose
is determined for each region. It is attempted to flnd the optimum dose distribution by
the general-purpose optimization scheme, Simulated Annealing. Through computer simu-
lation, the efiects of the factors such as dose distribution, total dose and developing time,
and performance of the dose control scheme have been analyzed. The future research efiorts
include application of the proposed methods to more general structures and multiple lines,
and experimental veriflcation.
45
Bibliography
[1] Ampere A. Tseng, Kuan Chen, Chii D. Chen, and Kung J. Ma, "Electron beam lithog-
raphy in nanoscale fabrication: recent development," IEEE Transactions on Electronics
Packaging Manufacturing vol. 26, no. 2, pp. 141-149, April 2003.
[2] W. Hu, K. Sarveswaran, M. Lieberman and G. H. Bernstein, "High-resolution electron
beam lithography and DNA nano-patterning for molecular QCA," IEEE Transactions
on In Nanotechnology, vol. 4(3), pp. 312-316, 2005.
[3] H. Kawano, H. Ito, K. Mizuno, T. Matsuzaka, K. Kawasaki, N. Saitou, H. Ohta, and Y.
Sohda, "Development of an electron-beam lithography system for high accuracy masks,"
Journal of Vacuum Science and Technology, B 21(2), pp. 823-827, Mar. 2003.
[4] S. Y. Chou etal., "Sub-10 nm imprint lithography and applications," Journal of Vacuum
Science and Technology, B 15(6), pp. 2897-2904, Nov./Dec. 1997.
[5] T. H. P. Chang, "Proximity efiect in electron-beam lithography," Journal of Vacuum
Science and Technology, 12(6), pp. 1271-1275, Nov./Dec. 1975.
[6] Sherry J. Gillespie, "Resist proflle control in e-beam lithography", IBM J. Res. Develop.,
pp. 454-460, July 1984.
[7] Y. C. Patil, A. Teolis, D. Park, R. Bass, K. Rhee, B. Bradie, M. C. Peckerar, "An error
measure for dose correction in e-beam nanolithography", Journal of Vacuum Science
and Technology, B 8(6), Nov./Dec. 1990.
[8] V. V. Aristov, B. N. Gaifullin, A. A. Svinstov, S. I. Zaitsev, H. F. Raith, R. Jede,
"Accuracy of proximity efiect correction in electron lithography after development",
Journal of Vacuum Science and Technology, B 10(6), Nov./Dec. 1992.
[9] Jianyun Zhou, Xiaomin Yang, "Monte Carlo simulation of process parameters in elec-
tron beam lithography for thick resist patterning", Journal of Vacuum Science and
Technology, B 24(3), May/June 2006.
[10] M. Parikh, "Correction to proximity efiects in electron-beam lithography. II. Implemen-
tation", Journal of Applied Physics, vol. 50, no. 6, pp. 4371-4382, June 1979.
[11] N. D. Wittels and C. Youngman, "Proximity efiect correction in electron-beam lithogra-
phy", Proceedings of the 8th International Conference on Electron and Ion Beam Science
and Technology, ed. by R. Bakish, Electrochemical Society, Princeton, New York, pp.
361-365, 1978.
46
[12] D. P. Kern, "A Novel Approach to Proximity Efiect Correction", Proceedings of the 9th
International Conference on Electron and Ion Beam Science and Technology, ed. by R.
Bakish, Electrochemical Society, Princeton, New York, pp. 326-339, 1980.
[13] D. Chow, J. McDonald, C. King, W. Smith and A. J. Steckl, "An image processing
approach to fast, e?cient proximity efiect correction for electron-beam lithography",
Journal of Vacuum Science and Technology, vol. 1, pp. 1383-1390, Oct./Dec. 1983.
[14] J. Greeneich, "Proximity efiect correction using a dose compensation curve", Journal
of Vacuum Science and Technology, vol. 19, pp. 1269-1274, Nov./Dec. 1983.
[15] M. Parikh, "Calculation of changes in pattern dimensions to compensate for proximity
efiects in electron-beam lithography", Journal of Applied Physics, vol. 51, pp. 705-709,
January 1980.
[16] M. Parikh, "Correction to proximity efiects in electron-beam lithography. III. Experi-
ments", Journal of Applied Physics, vol. 50, no. 6, pp. 4383-4387, June 1979.
[17] M. Parikh, "Proximity Efiect Corrections in Electron Beam Lithography", Proceedings
of the 8th International Conference on Electron and Ion Beam Science and Technology,
ed. by R. Bakish, Electrochemical Society, Princeton, New York, pp. 382-391, 1978.
[18] S.K.S. Ma, M. Parikh, and W. Ward, "Proximity Corrections in a Raster Scan Electron
Lithography Machine", Journal of Vacuum Science and Technology, vol. 19, no. 4, pp.
1275-1278, Nov./Dec. 1981.
[19] W. T. Lynch, T. E. Smith, W. Fichtner, "An algorithm for proximity efiect correction
with E beam exposure", International Conference on Microlithography, Microcircuit
Engineering, pp. 309-314, Oct. 5-8, 1982.
[20] G. Owen, "Methods for proximity efiect correction in electron-beam lithography", Jour-
nal of Vacuum Science and Technology, vol. 8, pp. 1889-1892, Nov./Dec. 1990.
[21] S.-Y. Lee, J.C. Jacob, C.M. Chen, J.A. McMillan, and N.C. MacDonald, "Proxim-
ity efiect correction in electron-beam lithography: A hierarchical, rule-based scheme-
PYRAMID", Journal of Vacuum Science and Technology, vol. B9, pp. 3048-3053,
Nov./Dec. 1991.
[22] J.C. Jacob, S.-Y. Lee, J.A. McMillan, and N.C. MacDonald, "Fast proximity efiect
correction: An extension of PYRAMID for circuit patterns of arbitrary size", Journal
of Vacuum Science and Technology, vol. B10, pp. 3077-3082, Nov./Dec. 1992.
[23] J.C. Jacob, "PYRAMID - A hierarchical, rule-based scheme for proximity efiect correc-
tion in electron-beam lithography: An Implementation for single layer circuits", Masters
Thesis, Cornell University, 1992.
[24] S.-Y. Lee and B. D. Cook, "PYRAMID-A Hierarchical, Rule-Based Approach Toward
Proximity Efiect Correction-Part I: Exposure Estimation," IEEE Transactions on Semi-
conductor Manufacture, vol. 11, no. 1, pp. 108-116, Feb. 1998.
47
[25] B. D. Cook and S.-Y. Lee, "PYRAMID-A Hierarchical, Rule-Based Approach Toward
Proximity Efiect Correction-Part II: Correction," IEEE Transactions on Semiconductor
Manufacture, vol. 11, no. 1, pp. 117-127, Feb. 1998.
[26] B. D. Cook, "PYRAMID - A Hierarchical, Rule-Based Proximity Efiect Correction
Scheme for electron-beam lithography: Generalization and optimization for homoge-
neous substrates", Ph.D. Dissertation, Cornell University, August 1996.
[27] B. D. Cook and S.-Y. Lee, "Dose, shape and hybrid modiflcations for PYRAMID in
electron beam proximity efiect correction", Journal of Vacuum Science and Technology,
B 15(6), pp. 2303-2308, Nov./Dec. 1997.
[28] B. D. Cook and S.-Y. Lee, "Fast Exposure Simulation for Large Circuit Patterns in
Electron Beam Lithography", Proceedings of the International Conference on Image
Processing, pp. 442-445, October 1995.
[29] B. D. Cook and S.-Y. Lee, "Performance Analysis and Optimal Parameter Selection for
PYRAMID", Proceedings of the International Conference on Mico- and Nano- Engi-
neering, pp. 61-64, September, 1995.
[30] S.-Y. Lee and B. D. Cook, "Interior Area Removal Method for PYRAMID", Journal of
Vacuum Science and Technology, vol. B 12(6), pp. 3449-3454, Nov./Dec. 1994.
[31] B. D. Cook and S.-Y. Lee, "A Quantitative Performance Analysis of PYRAMID", Pro-
ceedings of the International Conference on VLSI and CAD, pp. 16-18, November, 1993.
[32] B. D. Cook and S.-Y. Lee, "Fast Proximity Efiect Correction: An Extension of PYRA-
MID for Thicker Resists", Journal of Vacuum Science and Technology, vol. B 7(6), pp.
1556-1560, Nov./Dec. 1989.
[33] S.-Y. Lee, B. Liu, B. D. Cook, "Reducing recursive efiect for fast proximity correction",
Microelectronic Engineering, vol. 35, pp. 491-494, 1997.
[34] S.-Y. Lee and J. Laddha, "Adaptive selection of control points for improving accuracy
and speed of proximity efiect correction", Journal of Vacuum Science and Technology,
vol. B 16(6), pp. 3269-3274, Nov./Dec. 1998.
[35] J. Laddha and S.-Y. Lee, "Application of neural network to enhancing accuracy of E-
beam proximity efiect correction", Microelectronic Engineering, no. 53, pp. 317-320,
2000.
[36] S.-Y. Lee and D. He, "Simultaneous dose modiflcation for balanced e-beam proximity
efiect correction minimizing CD error", Microelectronic Engineering, no. 69, pp. 47-56,
2003.
[37] S.-Y. Lee and B. Liu, " Region-wise proximity efiect correction for heterogeneous sub-
strates in e-beam lithography:shape modiflcation", Journal of Vacuum Science and
Technology, B 14(6), pp. 3874-3879, Nov./Dec. 1996.
48
[38] F. Hu and S.-Y. Lee, "Dose control for fabrication of grayscale structures using a single
step electron-beam lithographic process", Journal of Vacuum Science and Technology,
B 21(6), pp. 2672-2679, Nov. 2003.
[39] S.-Y. Lee, F. Hu, and J. Ji, "Representation of non-rectangular features for exposure
estimation and proximity efiect correction in electron-beam lithography", Journal of
Vacuum Science and Technology, B 22(6), pp. 2929-2935, Nov./Dec. 2004.
[40] S.-Y. Lee and K. Anbumony, "Analysis of three-dimensional proximity efiect in electron-
beam lithography," Microelectronic Engineering, vol. 83(2), pp. 336-344, Feb. 2006.
[41] S.-Y. Lee and K. Anbumony, "Accurate control of remaining resist depth for nanoscale
three-dimensional structures in electron-beam grayscale lithography," Journal of Vac-
uum Science and Technology, B 25(6), pp. 2008-2012, Nov. 2007.
[42] K. Anbumony and S.-Y. Lee, "True three-dimensional proximity efiect correction in
electron-beam lithography," Journal of Vacuum Science and Technology, B 24(6), pp.
3115-3120, Nov./Dec. 2006.
[43] M. Gentili, L. Grella, A. Lucchesini, L. Luciani, L. Mastrogiacomo, and P. Musumeci,
"Energy density function determination in very-high-resolution electron-beam lithogra-
phy", Journal of Vacuum Science and Technology, B 8(6), pp. 1867-1871, Nov./Dec.
1990.
[44] M. Osawa, T. Yao, H. Aoyama, K. Ogino, H. Hoshino, Y. Machida, S. Asai, and H.
Arimoto, "Correction for local  are efiects approximated with double Gaussian proflle in
ArF lithography", Journal of Vacuum Science and Technology, B 21(6), pp. 2806-2809,
Nov./Dec. 2003.
[45] J. Kim, D. C. Joy and S.-Y. Lee, "Controlling resist thickness and etch depth for
fabrication of 3D structures in electron-beam grayscale lithography," Microelectronic
Engineering, vol. 84(12), pp. 2859-2864, Dec. 2006.
[46] D. Kyser and N. Viswanathan, "Monte Carlo simulation of spatially distributed beams
in electron-beam lithography," Journal of Vacuum Science and Technology, vol. 12(6),
pp. 1305-1308, Nov. 1975.
[47] Y. Hirai, S. Tomida, K. Ikeda, M. Sasago, M. Endo, S. Hayama, and N. Nomura,
"Three-dimensional resist process simulator PEACE (Photo and electron beam lithog-
raphyanalyzingcomputerengineeringsystem),"IEEE Transactions on Computer-Aided
Design vol. 10, pp. 802-807, June 1991.
[48] P. Li and S.-Y. Lee, "Step-width adjustment in fabrication of staircase structures,"
Journal of Vacuum Science and Technology, B 28(1), pp. 30-35 Jan. 2010.
[49] W. Gofie, G. Ferrier, and J. Rogers, "Global optimization of statistical functions with
simulated annealing," Journal of Econometrics, vol. 60, pp. 65-99, Jan./Feb. 1994.
49