Microelectronic Engineering 133 (2015) 78–87
Contents lists available at ScienceDirect
Microelectronic Engineering
journal homepage: www.elsevier.com/locate/mee
Dependency analysis of line edge roughness in electron-beam
lithography
X. Zhao a, S.-Y. Lee a,⇑, J. Choi b, S.-H. Lee b, I.-K. Shin b, C.-U. Jeon b, B.-G. Kim b, H.-K. Cho b
a
b
Department of Electrical and Computer Engineering, Auburn University, Auburn, AL 36849, United States
Samsung Electronics, Mask Development Team, 16 Banwol-Dong, Hwasung, Kyunggi-Do, Republic of Korea
a r t i c l e
i n f o
Article history:
Received 24 August 2014
Accepted 21 November 2014
Available online 5 December 2014
Keywords:
Exposure fluctuation
Line edge roughness
Monte Carlo simulation
Stochastic PSF
a b s t r a c t
The line edge roughness (LER) has become one of the critical issues which affect the minimum feature
size and the maximum circuit density realizable in most lithographic processes. Since the LER does not
scale with the feature size, it needs to be minimized as the feature size is reduced well below 100 nm.
One of the main factors contributing to the LER is the stochastic fluctuation of exposure. In the past, most
of the LER researches were based on a 2-D model without considering the resist depth dimension. In this
study, the dependency of the LER, caused by the stochastic fluctuation of exposure due to electron scattering in the resist and the shot noise due to variation of electron influx, on lithographic parameters such
as shot noise, beam energy, exposing interval, dose, etc., has been investigated with a 3-D model, as the
first step toward developing an effective method for minimizing the LER. In the case of CAR, the effect of
developing process on LER is also considered. In this paper, the results from an extensive simulation are
reported with a detailed discussion.
Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction
Electron-beam (e-beam) lithography is one of the widely-used
methods for transferring patterns onto the resist layer [1–4]. Despite
the low throughput due to pixel-by-pixel or feature-by-feature writing and the proximity effect caused by electron scattering, its capability of being able to write ultra-fine features has a variety of
applications such as fabrication of photo-masks, low-volume production of semiconductor components, experimental circuit patterns, etc. The well-known proximity effect, which causes
deviation from the target dimensions of a feature in the written pattern, has been studied over three decades. Many proximity correction schemes have been devised to minimize the critical
dimension (CD) error and eventually increase the circuit density,
i.e., dose modulation, pattern biasing, GHOST, etc. [5–8]. Another
related issue is the variation of CD within a feature due to the stochastic nature of lithographic and developing processes. A quantitative measure of such variation which is being extensively studied
these days is the line edge roughness (LER) [9]. Since the LER does
not scale with the feature size, it can significantly limit the minimum
feature size and maximum pattern density that can be achieved as
the feature size is reduced well below 100 nm [10,11]. Therefore, it
⇑ Corresponding author. Fax: +1 (334) 844 1809.
E-mail address: leesooy@eng.auburn.edu (S.-Y. Lee).
http://dx.doi.org/10.1016/j.mee.2014.11.017
0167-9317/Ó 2014 Elsevier B.V. All rights reserved.
is unavoidable to address the issue of LER in order to be able to continue to shrink the feature size and minimize the malfunctioning of a
device due to the LER.
There are several factors which contribute to the LER in e-beam
lithography. One of the major factors is the stochastic fluctuation
of exposure (energy deposited in the unit volume of resist), which
is caused by shot noise (variation of electron flux) and random
scattering of electrons. In order to develop an effective method to
reduce the LER, it is essential to analyze the characteristics of
LER. In the past, the LER was studied using a two-dimensional
(2-D) model in most cases, i.e., the resist depth dimension was
ignored. However, different layers of resist may exhibit different
behaviors of LER. In this study, a 3-D model of substrate system
is employed to thoroughly analyze the dependency of LER, caused
by the stochastic fluctuation of exposure, on factors such as edge
location, resist layer, resist thickness, etc. Also, the e-beam lithographic parameters which affect the stochastic fluctuation of exposure and therefore the LER are identified, e.g., shot noise, dose (the
amount of charge given to each unit area of resist surface), beam
energy, beam diameter, and exposing interval, and their effects
on the LER are analyzed with the 3-D model. The type of resist
may also have a substantial effect on the LER. Two different types
of resists, PMMA (poly(methyl methacrylate)) and chemically
amplified resist (CAR) PHS (poly(4-hydroxystyrene)), are considered. In the case of CAR, the effect of the randomness involved in
X. Zhao et al. / Microelectronic Engineering 133 (2015) 78–87
79
the resist developing process is also analyzed. It needs to be
pointed out that the main focus of this study is on understanding
the behavioral trend (not the absolute level) of LER as those factors
and parameters vary.
The rest of the paper is organized as follows. The model of the
LER simulation is introduced in Section 2. The detail of simulation
procedures is described in Section 3. Simulation results are discussed in Section 4, followed by a summary in Section 5.
2. Modeling of LER
Modeling the LER may be done analytically or via simulation. In
this study, a simulation approach is taken mainly for its flexibility
at the expense of high computational requirement. The stochastic
exposure distribution in the resist is computed, and the exposure
is converted into the developing rate (a quantitative measure of
how fast resist is developed) point-by-point. Then, the 3-D remaining resist profile is obtained through simulation of resist development. From the resist profile, the boundaries of a feature are
determined on each layer of resist and the LER is quantified by a
certain measure, e.g., the standard deviation of edge location.
2.1. Exposure distribution
A point spread function (PSF), psf ðx; y; zÞ, describes the spatial
distribution of exposure throughout the resist when a single point
is exposed. Due to the random nature of electron scattering and
shot noise, the PSF is stochastic, i.e., psf ðx; y; zÞ is random at each
point ðx; y; zÞ. An instance of stochastic PSF is shown in Fig. 1. Since
the PSF is stochastic, the exposure distribution is accordingly
stochastic.
The stochastic exposure distribution may be obtained by
employing the Monte Carlo simulation at each point exposed by
the e-beam, which is equivalent to generating an instance of stochastic PSF for each point exposed. This approach may lead to a
more realistic exposure distribution. However, its computational
complexity of generating the PSF’s is too high to be practical for
most patterns of realistic size. Hence, a new method to greatly
reduce the number of stochastic PSF’s (instances) to be generated,
referred to as the simplified Monte Carlo simulation (SMC) method,
was recently developed [12]. It generates only a small number of
stochastic PSF’s and selects a PSF randomly for each point exposed
for calculation of the exposure distribution. Note that we are
mainly interested in certain measures of the exposure fluctuation,
not the exact distribution of exposure itself. Through simulation, it
was shown that the SMC method is able to generate exposure distributions statistically equivalent to those by the direct Monte Carlo method. Therefore, in this study, the SMC method is employed
without compromising the accuracy of estimating the LER.
A typical substrate system is shown in Fig. 2 where the X–Y
plane corresponds to the surface of resist and the Z-axis is along
the resist depth dimension. The exposure distribution eðx; y; zÞ in
the resist for a feature is computed by the convolution between
the stochastic PSF psf ðx; y; zÞ and dose distribution function
dðx; y; 0Þ defined on the surface of resist,
eðx; y; zÞ ¼
ZZ
0
dðx x0 ; y y0 ; 0Þ psf ðx0 ; y0 ; zÞdx dy
0
Fig. 1. An instance of the stochastic PSF at (a) top, (b) middle, and (c) bottom layers:
40 electrons per shot with exposing interval of 1 nm (640 lC=cm2 ), 300 nm PMMA
on Si, 50 keV and beam diameter of 3 nm.
ð1Þ
In the case of a uniform dose distribution, dðx; y; 0Þ can be expressed
as,
dðx; y; 0Þ ¼
D for exposed points
0
otherwise
ð2Þ
Fig. 2. A 3-D model of substrate system.
80
X. Zhao et al. / Microelectronic Engineering 133 (2015) 78–87
2.2. Resist profile and LER
In order to derive the (remaining) resist profile from the exposure distribution in the resist, a procedure of simulating the resist
development process is employed. For the simulation, how fast
each point in the resist is developed, i.e., developing rate, needs
to be known. Note that the exposure itself does not provide a quantitative measure of developing rate. In determining the relationship between the exposure and developing rate, an experimental
approach is taken in this study. That is, the conversion formula
which maps each exposure level to the corresponding developing
rate is derived based on experimental results by matching the simulated resist profile to the experimentally-obtained resist profile.
Such a formula would reflect the characteristics of a developer
and developing process. The exposure at each point in the resist
is converted into the developing rate according to the conversion
formula.
A common drawback of the conventional resist development
simulation methods such as the cell removal method [13] is that
they are computationally intensive. Therefore, it is not practical
to employ them in an extensive simulation study. This time-consuming nature is mainly due to the incremental updating of cell
states through a large number of iterations. In order to overcome
the drawback, a new fast method for resist development simulation was developed [14]. The method takes a path-based approach,
i.e., the resist development process is modeled by development
paths starting from the resist surface toward the boundaries of
(remaining) resist profile. Each path consists of a vertical path segment (representing vertical development) and lateral path segments (representing lateral development). All possible paths are
traced without iterations, and those paths reaching the farthest
points, given a developing time, determine the boundaries of resist
profile. It has been shown that this path-based method generates
resist profiles well matched with those by the cell removal and fast
marching methods and greatly reduces the simulation time, especially for features of regular shapes such as lines, rectangles, circles,
etc.
The 3-D profile of remaining resist of a feature is obtained by
the path-based method. From the 3-D profile, the feature boundaries, i.e., edges, at each layer of resist are detected (see Fig. 3).
Referring to Fig. 3, let xe ðy; ZÞ denote the actual edge location along
one side of a feature (e.g., a long line) at a resist layer with z ¼ Z. In
this study, the LER is quantified at each layer as 3 times the stan-
dard deviation of edge location, i.e., 3r-LER, along the length
dimension of the feature. That is, the LER at the layer with z ¼ Z
may be expressed as follows.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
Z
1
2
ðxe ðy; ZÞ xe ðy; ZÞÞ dy
LER ¼ 3
N
ð3Þ
R
where xe ðy; ZÞ ¼ N1 xe ðy; ZÞdy and N is the length of feature segment
over which the LER is evaluated.
3. Simulation
The substrate system for the simulation is composed of a resist
on Si. Two different types of resist, PMMA and CAR, are considered
in simulation and analysis. The resist is partitioned into five layers,
and the top, middle and bottom layers are considered in the simulation. The 3-D exposure distribution in the resist for a feature of
size W L is computed layer-by-layer within a window of size
M N as shown in Fig. 4. Through the resist development simulation, the remaining resist profile is obtained on each layer. Then,
the LER is evaluated along the length dimension of a feature. In
order to avoid a biased result, the simulation is repeated five times
to get the averaged LER.
The default values of the parameters are as follows. For PMMA,
the thickness of the resist is 300 nm, the beam energy is 50 keV,
the beam diameter is 3 nm, the exposing interval is 1 nm, and
the dose is 640 lC/cm2. For CAR, the resist thickness is 300 nm,
the beam energy is 50 keV, the beam diameter is 3 nm, the exposing interval is 1 nm, and the dose is 32 lC/cm2. Unless specified
otherwise, these default values are used in the simulation. In the
dependency analysis, each of these parameters is varied.
The coordinate system is set up such that the X-axis is perpendicular to the edge of line feature and X ¼ 0 corresponds to the target edge location as can be seen in Figs. 3 and 4. The LER is
examined with the (actual) edge location varied from the inside
(X < 0) of a feature to the outside (X > 0) by controlling the developing time. The dependency of the LER on the parameters is analyzed mostly for a single feature. However, in order to study the
dependency of the LER on location within a large pattern, a pattern
of multiple lines is also employed as shown in Fig. 5. Two regions,
center and corner, within the pattern are considered.
In the remainder of this section, each of the lithographic parameters including shot noise to be considered in the simulation is
described.
xe( y, Z)
Fig. 3. Dependency of the LER on edge location, from the inside (X < 0) of the
feature to the outside (X > 0), is analyzed by controlling the developing time. The
edge location of X ¼ 0 corresponds to the target edge location. xe ðy; ZÞ denotes the
feature edge (boundary), after resist development, along its length dimension (y) at
a certain resist layer with z ¼ Z.
Fig. 4. The exposure is computed within an exposure window in each layer, which
is partially overlapped with a feature so that the exposure fluctuation is analyzed in
both exposed and unexposed regions.
X. Zhao et al. / Microelectronic Engineering 133 (2015) 78–87
81
number of electrons per shot. For a higher beam energy, the number of electrons per shot needs to be increased, resulting in a higher
dose, to maintain the same exposure level. That is, dependency of
the LER on beam energy is analyzed under the constraint that the
exposure level is kept constant by varying the dose level. Since the
exposure level varies along the resist depth dimension, the average
exposure over layers is used for the equalization of exposure level.
3.3. Resist thickness
Fig. 5. Dependency of the LER on location within a large pattern. Two regions,
center and corner, are considered.
3.1. Shot noise
The effect of shot noise on the LER was studied by many
researchers [15–19]. A given dose level determines the number
of electrons to be given to each point, which is to be the same
for all points in the ideal case. However, the actual number of electrons varies with point and shot noise refers to this variation. Shot
noise causes an additional fluctuation to the exposure distribution
which is stochastic due to random scattering of electrons
(described by stochastic PSF’s). Let N 0 denote the number of electrons to be given to each point, corresponding to a specified dose.
The actual number of electrons is known to follow the Poisson dispffiffiffiffiffiffi
tribution with the standard deviation of N 0 . Thus, the signal to
(shot) noise ratio (SNR) is given by
pffiffiffiffiffiffi
N0
SNR ¼ pffiffiffiffiffiffi ¼ N0
N0
ð4Þ
That is, the lower the dose (N0 ) is, the smaller the SNR is, i.e., leading
to a larger additional fluctuation of exposure distribution. In this
study, when effects of other lithographic parameters are analyzed,
shot noise is turned off (not included). This is to clearly understand
the effect of each parameter only, not affected by shot noise, which
will be helpful to the future effort of developing a method of LER
minimization.
A set of random integers following the Poisson distribution with
the mean of N 0 is generated. Each time an instance of stochastic
PSF is generated through the Monte Carlo simulation, an integer
from the set is selected as the number of electrons to be simulated
(traced). Note that these PSF’s are used in computing the exposure
distribution in our SMC method (refer to Section 2).
3.2. Beam energy
Three different levels of beam energy, 25, 50, and 75 keV, are
considered. When the beam energy is higher, most of the electrons
go through the resist depositing less energy in the resist, leading to
a lower exposure level. In order to extract the dependency of stochastic exposure fluctuation on the beam energy only (excluding
the dependency on the exposure level), the exposure level is maintained at the same level for all levels of beam energy by adjusting
the dose according to the beam energy level. With the exposing
interval (see Section 3.5) fixed, the dose level is equivalent to the
At an upper layer of resist (close to the surface of resist), electrons have not yet experienced scattering much. Therefore, the stochastic fluctuation of exposure is small and the exposure contrast
over the edge of feature is high. As electrons travel deeper into
the resist, they experience more scattering, leading to a larger stochastic fluctuation of exposure. Also, the increased level of scattering makes the main peak of PSF broader and therefore a lower
exposure contrast over the feature edge. That is, at a lower layer
(close to the substrate), the exposure contrast over the feature
edge is smaller and the stochastic fluctuation of exposure is larger.
These differences between the upper and lower layers of resist
become larger for a thicker resist. Hence, dependency of LER on
resist thickness is analyzed at the bottom layer where the LER
shows the maximum difference for different thicknesses of resist.
Three different resist thicknesses, 100, 300, and 500 nm, are
considered.
3.4. Beam diameter
Given a number of electrons per shot (i.e., a dose), the beam
diameter affects the density of electrons entering the resist and
accordingly the shape of PSF, especially the main peak. For example, as the beam diameter increases, electrons spread over a larger
area of beam cross-section and accordingly enter a larger (circular)
surface area of resist. This makes the spatial distribution of exposure near the point exposed (by such a beam) wider and therefore
the main lobe of PSF broader, in turn, the spatial distribution of
exposure smoother. Note that the main lobe of PSF represents
the forward scattering of electrons. The effect of beam diameter
on the LER is investigated by changing the beam diameter from 1
to 10 nm.
3.5. Exposing interval
When a Gaussian beam is used in an e-beam system, i.e., the
resist is exposed point-by-point, the exposing interval, i.e., the distance between adjacent points exposed by the beam, has two different effects. It affects the periodic variation of (spatial) exposure
distribution and the number of electrons per shot (point). Unless
the exposing interval is very large, the spatial distribution of exposure is contiguous due to the electron scattering (i.e., the PSF is not
a delta function). However, it may show a periodic variation
depending on the exposing interval. In general, a longer exposing
interval makes the periodic variation of exposure larger. On the
other hand, for a longer exposing interval, the number of electrons
per shot needs to be increased for the same dose level, which
improves (increases) the SNR, i.e., a smaller stochastic fluctuation
of exposure. That is, the exposing interval has two opposite effects
and therefore it would be worthwhile to examine the tradeoff
between these two effects. For analyzing the dependency of LER
on exposing interval, three different exposing intervals, E; 2E and
4E, are considered as illustrated in Fig. 6 where E is 1 nm. To maintain the same area dose, the number of electrons per shot is set
proportional to the square of exposing interval. In all cases, the
pixel interval, i.e., the distance between adjacent points at which
exposure is computed in simulation, is set to E.
82
X. Zhao et al. / Microelectronic Engineering 133 (2015) 78–87
rðx; y; zÞ ¼ ðRmax Rmin Þ
a¼
ða þ 1Þð1 mðx; y; zÞÞn
þ Rmin
a þ ð1 mðx; yzÞÞn
nþ1
ð1 mth Þn
n1
ð6Þ
ð7Þ
where, Rmax ; Rmin , and n are the maximum developing rate, the minimum developing rate, and the developer contrast, respectively.
In the above expressions, mth is a threshold of mðx; y; zÞ corresponding to the inflection point of the developing rate curve, and
n and mth are computed as
mth ¼ 1 Fig. 6. Dependency of the LER on the exposing interval is analyzed considering
three different intervals, where the smallest exposing interval E is 1.0 nm.
3.6. Dose level
A different dose corresponds to a different number of electrons
per shot (with the exposing interval fixed). The number of electrons
given to a point directly affects the statistical property of exposure.
The effect of dose on the LER is studied by changing the dose level
from 160 lC/cm2 to 1280 lC/cm2 for PMMA and from 16 lC/cm2 to
64 lC/cm2 for CAR (the typical dose level required for CAR is much
lower than that for PMMA). As the dose level changes, the exposure
level changes proportionally and therefore the developing time is
adjusted accordingly so that the LER at the same edge location
can be compared among different doses.
k1
P
n ¼ kb1 þ 0:08385ð1 kmth Þc
ð8Þ
ð9Þ
As can be seen in the above equations, P and k also determine the
developing-rate contrast which affects the LER. The dependency of
LER on P and k is studied by changing P from 10 to 30 with k fixed
at 6 and changing k from 3 to 17 with P fixed at 20.
In the simulation of CAR, unless specified otherwise, the average PAG density is set to 0:05 nm3 , P to 13, k to 6, Rmax to
55 nm=sec, and Rmin to 0:15 nm=sec.
4. Results and discussion
The CASINO software [22] is employed to generate stochastic
PSF’s assuming a Gaussian electron beam. The single line feature
considered in the simulation is 25 500 nm2 and the exposure
window is 45 320 nm2 . In the direct Monte Carlo simulation,
12500 PSF’s would be required for the simulation with this single
feature. However, thanks to the SMC method, only 50 PSF’s need
3.7. CAR
In the case of CAR, the effect of the developing process including
post-exposure-bake (PEB) on the LER can be relatively significant,
compared to a non-CAR resist like PMMA. Therefore, not only the
effect of exposure fluctuation but also that of the stochastic processes in resist development is analyzed in this study.
During the exposure process, photoacid generators (PAG)
decompose generating acid. The acid concentration hðx; y; zÞ can
be expressed by
hðx; y; zÞ ¼ PAG0 ðx; y; zÞð1 eCeðx;y;zÞ Þ
(a)
ð5Þ
where C is the exposure rate constant, and PAG0 ðx; y; zÞ is the number of PAG’s in each cell. PAG0 ðx; y; zÞ fluctuates following a Poisson
distribution, which causes the fluctuation in acid concentration
contributing to the LER [20].
The effect of fluctuation of acid concentration on the LER is
studied by changing the PAG density from 0.05 to 0.4 nm3.
A PEB process is employed to thermally induce a chemical
reaction of PAG [21]. Consider a resist made up of phenolic polymers each with P phenol groups, some of which are initially
blocked (protected). The acid generated from the PAG catalyzes
the deblocking (deprotection) process of these phenol groups
during PEB. A polymer molecule becomes soluble when at least
a certain number (to be denoted by k) of phenol groups are
deblocked. The probability of that a polymer molecule is soluble
depends on P; k, and the probability that a phenol group is
deblocked (ionized). That is, the solubility is stochastic contributing to the LER.
According to the Mack’s dissolution model [21], the developing
rate rðx; y; zÞ at each point in the resist can be expressed as a function of the corresponding normalized concentration of unreacted
blocking phenol groups, mðx; y; zÞ:
(b)
Fig. 7. Layer dependency of the LER: (a) without shot noise and (b) with shot noise.
Dose: 640 lC=cm2 , resist thickness: 300 nm, beam energy: 50 keV, beam diameter:
3 nm, and exposing interval: 1 nm.
X. Zhao et al. / Microelectronic Engineering 133 (2015) 78–87
83
(a)
Fig. 8. Fluctuation of exposure measured on different layers with the default
substrate system.
(b)
(a)
(b)
Fig. 9. Dose dependency of the LER: (a) without shot noise and (b) with shot noise.
Resist thickness: 300 nm, beam energy: 50 keV, beam diameter: 3 nm, exposing
interval: 1 nm, bottom layer.
to be generated. In the multiple-line pattern, there are 41 lines
where each line is 25 2000 nm2 and the space between lines is
25 nm. The exposure window of 45 160 nm2 is placed at the center and corner regions.
The results for PMMA are provided in Figs. 7–13, and those for
CAR in Figs. 14–18.
4.1. Edge location and layers
In Fig. 7a, it is seen that, as the edge location is moved from the
inside of the feature to the outside, the LER drops down quickly
and then becomes almost flat. This is mainly due to the facts that
the (absolute) fluctuation of exposure decreases fast from the
exposed area to the unexposed area and changes only slightly as
Fig. 10. (a) Beam energy dependency of the LER with shot noise on bottom layer.
Dose: 400, 640, and 880 lC/cm2 for beam energy of 25, 50, and 75 keV, respectively,
resist thickness: 300 nm, beam diameter: 3 nm, and exposing interval: 1 nm. (b)
Resist thickness dependency with shot noise on bottom layer. Dose: 640 lC=cm2 ,
Beam energy: 50 keV, beam diameter: 3 nm, and exposing interval: 1 nm.
we go farther into the unexposed area as shown in Fig. 8 and that
the exposure contrast is large over the feature edge (a larger exposure contrast tends to make the LER smaller). It can also be understood from this figure that a significant variation of developing
time may change the LER greatly since it changes the edge location.
It is also observed that the LER around the target edge location
is substantially larger at a lower layer of resist. At a lower layer,
more scattering of electrons occurs over a larger space, leading to
a larger fluctuation of exposure and a smaller exposure contrast
over the feature edge. In addition, it needs to be noted that the
LER difference between the inside and outside of a feature is larger
at an upper layer. This is because the exposure contrast and therefore the contrast of exposure fluctuation over the feature edge are
larger at an upper layer.
4.2. Shot noise
As expected and shown in Fig. 7b, shot noise increases the LER
substantially at all three layers due to the increased fluctuation of
exposure. The effects of shot noise on different layers are also compared quantitatively in terms of the percentage increase of LER
defined as
jpðiÞ qðiÞj
100%
pðiÞ
ð10Þ
where pðiÞ is the LER at the edge location i without shot noise and
qðiÞ is the LER at the same edge location with shot noise. Table 1
provides the percentage increase of LER at five different edge locations. It is seen that the percentage increase is larger at a lower
84
X. Zhao et al. / Microelectronic Engineering 133 (2015) 78–87
(a)
(a)
(b)
(b)
Fig. 11. (a) Beam diameter dependency of the LER on bottom layer. Dose: 640 lC/
cm2, resist thickness: 300 nm, and beam energy: 50 keV. (b) Exposing interval
dependency on bottom layer. Dose: 160 lC=cm2 . Resist thickness: 300 nm, beam
energy: 50 keV, and beam diameter: 3 nm.
Fig. 12. Location dependency of the LER in a large pattern: (a) top layer and (b)
bottom layer. Dose: 640 lC=cm2 , beam energy: 50 keV, resist thickness: 100 nm,
beam diameter: 3 nm, and exposing interval: 1 nm.
(a)
layer. The effect of shot noise on the LER is amplified at a lower
layer where the exposure fluctuation is larger compared to the
upper layers.
4.3. Dose level
In Fig. 9a, the LER is compared under different dose levels. A
higher dose contains a larger number of electrons per shot. Therefore, with a higher dose, the exposure is averaged over more electrons leading to a better statistics (a larger SNR), i.e., a smaller
fluctuation of exposure and accordingly a smaller LER. Also, the
effect of shot noise on the LER is examined under different dose
levels in Fig. 9b. It is seen that the LER is affected less by shot noise
when the dose level is higher, since the SNR (Eq. (4)) is higher for a
higher dose level.
(b)
4.4. Beam energy
As shown in Fig. 10a, the LER decreases with the increasing
beam energy. With a higher beam energy, electrons deposit their
energy less in the resist and therefore the dose level needs to be
increased for the same exposure level. This results in a higher
SNR leading to a smaller LER. When the beam energy is increased
from 25 to 50 keV, the LER decreases greatly. However, the
decrease in LER is significantly less when the beam energy is
increased from 50 keV to 75 keV. This is due to the fact that the
change in the scattering characteristics is much less as the beam
energy is increased beyond 50 keV.
Fig. 13. Location dependency of the LER in a large pattern: (a) top layer and (b)
bottom layer. Dose: 1280 lC=cm2 , Beam energy: 20 keV, resist thickness: 500 nm,
beam diameter: 3 nm, and exposing interval: 1 nm.
X. Zhao et al. / Microelectronic Engineering 133 (2015) 78–87
(a)
(a)
(b)
(b)
Fig. 14. Layer dependency of the LER (CAR): (a) without shot noise and (b) with
shot noise. Dose: 32 lC=cm2 , resist thickness: 300 nm, beam energy: 50 keV, beam
diameter: 3 nm, and exposing interval: 1 nm.
85
Fig. 15. Dose dependency of the LER (CAR): (a) without shot noise and (b) with shot
noise. Resist thickness: 300 nm, beam energy: 50 keV, beam diameter: 3 nm,
exposing interval: 1 nm, bottom layer.
4.5. Resist thickness
In Fig. 10b, the LER at the bottom layer of resist is plotted as a
function of the resist thickness. As discussed in Section 3.3, in a
thicker resist, there is a larger space of resist through which electrons scatter, increasing the stochastic fluctuation of exposure distribution and decreasing the exposure contrast over the feature
edge at a lower layer. The increased fluctuation and decreased contrast make the LER larger. On the other hand, in the case of thinner
resist, the stochastic exposure fluctuation stays small and the
exposure contrast remains high at the bottom layer. Therefore,
the LER at the bottom layer is smaller, compared to a thicker resist.
Also, most of electron energy is deposited in the resist through the
forward scattering of electrons. In a thinner resist, the forward
scattering is less and therefore the exposure level is lower for the
same dose level. This makes the absolute fluctuation of exposure
smaller. Hence, in general, the LER at the bottom layer (or the
LER averaged over layers) increases as the resist thickness
increases as can be seen in Fig. 10b.
Fig. 16. Dependency of the LER on PAG density (CAR). Resist thickness: 300 nm,
beam energy: 50 keV, beam diameter: 3 nm, exposing interval: 1 nm, dose:
32 lC=cm2 , P: 13, k: 6, bottom layer.
overlap and such periodic variation quickly diminishes. Therefore,
the LER stays almost unchanged beyond 3 nm of beam diameter.
4.6. Beam diameter
As shown in Fig. 11a, the LER is reduced substantially as the
beam diameter is increased from 1 to 3 nm. This is mainly due to
the fact that a larger beam diameter makes the forward scattering
part of PSF broader (refer to Section 3.4) and therefore the periodic
variation of exposure smaller. Also, the broader PSF decreases the
exposure contrast over the edge of feature, which tends to reduce
the LER. However, once the beam diameter becomes larger than
the exposing interval, the beams for two adjacent points start to
4.7. Exposing interval
As discussed in Section 3.5, changing the exposing interval has
two opposite effects on the LER. A smaller exposing interval
reduces the periodic variation of exposure, which tends to make
the LER smaller, but increases the randomness of PSF (because of
a smaller number of electrons per shot leading to a lower SNR),
which would increase the LER. In Fig. 11b, the LER is plotted as a
86
X. Zhao et al. / Microelectronic Engineering 133 (2015) 78–87
(a)
60
(b)
P = 30
P = 20
P = 10
50
40
30
20
10
0
0
0.2
0.4
0.6
0.8
1
m
Fig. 17. (a) Dependency of the LER on P (CAR) and (b) developing rate curve for
different P, where m is the (average) normalized concentration of unreacted
blocking phenol groups. Resist thickness: 300 nm, beam energy: 50 keV, beam
diameter: 3 nm, exposing interval: 1 nm, dose: 32 lC=cm2 , k: 6, average density of
PAG: 0:05 nm3 , bottom layer.
function of the exposing interval. It is seen that a smaller exposing
interval leads to a smaller LER. This suggests that the reduced periodic variation of exposure has a larger effect on the LER than the
increased randomness of PSF in the cases considered in this
simulation.
4.8. Multiple features
Developing Rate [nm / sec]
Developing Rate [nm / sec]
(b)
(a)
60
50
40
30
20
10
0
0
4.9. CAR
In Figs. 14 and 15, two sets of the simulation results obtained
for CAR are provided for comparison with PMMA. Referring to
0.2
0.4
0.6
0.8
1
m
Fig. 18. (a) Dependency of the LER on k (CAR) and (b) developing rate curve for
different k, where m is the (average) normalized concentration of unreacted
blocking phenol groups. Resist thickness: 300 nm, beam energy: 50 keV, beam
diameter: 3 nm, exposing interval: 1 nm, dose: 32 lC=cm2 , P: 20, average density of
PAG: 0:05 nm3 , bottom layer.
Table 1
Percentage increase of the LER due to the added shot noise.
Layer
In Figs. 12 and 13, the LER is compared between center and corner regions in the multiple-line pattern. In a large pattern, more
exposed points make exposure contributions to the center region,
compared to the corner region. Therefore, the background exposure is higher in the center region, leading to a lower exposure contrast over the feature edge. A lower exposure contrast tends to
make the LER larger. On the other hand, the stochastic fluctuation
of exposure is smaller in the center region since a larger number of
exposure contributions are averaged (added). The lower exposure
fluctuation makes the LER smaller. In which region the LER is larger
depends on a combination of these two factors, more precisely
their differences between the two regions.
In the case where the difference of exposure contrast is small
and the difference of exposure fluctuation is large, the LER is smaller in the center region as shown in Fig. 12. But, when the difference of exposure contrast is large and the difference of exposure
fluctuation is small, the LER is smaller in the corner region as
shown in Fig. 13.
k = 17
k = 10
k=3
Top
Middle
Bottom
Edge Location
3 (%)
1 (%)
0 (%)
+1 (%)
+3 (%)
28.4
50.9
30.0
34.5
40.0
41.3
121.2
119.1
124.7
21.1
28.3
28.9
56.5
100.0
76.2
the respective results for PMMA in Figs. 7 and 9, it is clear that
the absolute level of LER is substantially higher for CAR than for
PMMA. This is mainly due to the lower dose level required for
CAR and the added randomness involved in the developing process
of CAR. If the dose level increases, the LER would decrease as discussed earlier (Section 4.3). From the figures, it is also observed
that the behaviors of LER, e.g., dependency on edge location, resist
layer, shot noise and dose level, are similar with those observed for
PMMA earlier in this section.
In Figs. 16–18, the results from analyzing the effects of stochastic developing process of CAR on the LER in detail are provided.
Specifically, dependency of LER on each of the PAG density, the
number of phenol groups (P) per polymer, and the minimum number of phenol groups required for the polymer being soluble (k),
has been studied. With a higher PAG density, there are more PAG’s
per cell leading to a higher signal-to-noise ratio (note that the
number of PAG’s per cell follows a Poisson distribution). This in
X. Zhao et al. / Microelectronic Engineering 133 (2015) 78–87
turn leads to a smaller fluctuation of acid concentration and therefore a lower LER as shown in Fig. 16. However, when the PAG density continues to increase beyond a certain level, the LER starts to
increase. This may be explained by the fact that the relative contrast of developing rate initially increases and then decreases. As
discussed earlier (Section 4.8), a lower rate (exposure) contrast
tends to increase the LER. As P increases or k decreases, the developing-rate contrast increases as seen in Fig. 17b and 18b. This
increasing contrast of developing rate is the main factor which
makes the LER smaller. These behaviors can be observed in
Figs. 17a and 18a. Also, it is worthwhile to notice the dependency
of LER on P around the target edge location, i.e., as P increases, the
LER decreases quickly and then does not change significantly.
87
of phenol groups, or a smaller number of phenol groups required
for the polymer being soluble, the LER is smaller.
The results from this study must be helpful in understanding
the characteristics of the LER caused by the stochastic fluctuation
of exposure and stochastic developing process. The focus of our
current work is on reducing the LER based on the simulation
results.
Acknowledgement
This work was supported by a research grant from Samsung
Electronics Co., Ltd.
References
5. Summary
As the feature size in a pattern continues to be reduced, it is
unavoidable to consider and minimize the LER in maximize the
feature density and fabrication yield. This issue becomes increasingly important since the LER does not scale with feature size. In
this study, as the first step toward developing an effective scheme
for minimizing the LER, the dependency of LER, caused by the stochastic exposure, on various parameters has been analyzed
through an extensive simulation with a 3-D model. In simulation,
shot noise is also taken into account.
From the simulation results, the following observations can be
made. (i) The LER is substantially larger at a lower layer of resist.
(ii) The LER decreases as the beam energy (with the same exposure
level maintained) or dose increases. (iii) As the edge location is
moved from the inside of feature to the outside, the LER decreases.
(iv) The LER becomes smaller for a thinner resist. (v) The LER is larger for a larger exposing interval. (vi) The LER tends to be smaller
for a larger beam diameter. (vii) Shot noise has significant effect on
the LER and its effect is less for a higher dose or an upper layer of
resist. (viii) In a large pattern, the region-dependency of LER is
affected by both the contrast and stochastic fluctuation of
exposure.
CAR typically requires a lower dose level, compared to PMMA,
and therefore tends to suffer from a larger LER, however, exhibits
the similar behaviors of LER. A higher PAG density, a larger number
[1] R. Murali, D. Brown, K. Martin, J. Meindl, J. Vac. Sci. Technol. B 24 (2006) 2936.
[2] F. Hu, S.-Y. Lee, J. Vac. Sci. Technol. B 21 (2003) 2672.
[3] Q. Dai, S.-Y. Lee, S.-H. Lee, B.-G. Kim, H.-K. Cho, J. Vac. Sci. Technol. B 29 (2011)
06F314.
[4] Q. Dai, S.-Y. Lee, S.-H. Lee, B.-G. Kim, H.-K. Cho, Microelectron. Eng. 88 (2011)
902.
[5] A. Schleunitz, H. Schift, J. Micromech. Microeng. 20 (2010) 095002.
[6] E. Seo, B.K. Choi, O. Kim, Microelectron. Eng. 53 (2000) 305.
[7] S.-Y. Lee, B.D. Cook, IEEE Trans. Semicond. Manuf. 11 (1998) 117.
[8] G. Owen, P. Rissman, J. Appl. Phys. 54, 3573.
[9] C.M. Nelson, S.C. Palmateer, T. Lyszczarz, Proc. SPIE 3332, 1998.
[10] J. Thiault, J. Foucher, J.H. Tortai, O. Joubert, S. Landis, S. Pauliac, J. Vac. Sci.
Technol. B 23 (2005) 3075.
[11] L.H.A. Leunissen, W. Zhang, W. Wu, S.H. Brongersma, J. Vac. Sci. Technol. B 24
(2006) 1859.
[12] X. Zhao, S.-Y. Lee, S.-H. Lee, B.-G. Kim, H.-K. Cho, J. Vac. Sci. Technol. B 30
(2012) 06F308.
[13] Y. Hirai, S. Tomida, K. Ikeda, M. Sasago, M. Endo, S. Hayama, N. Nomura, IEEE
Trans. Comput. Aided Des. 10 (1991) 802.
[14] Q. Dai, R.- Guo, S.-Y. Lee, J. Choi, S.-H. Lee, I.-K. Shin, C.-W. Jeon, B.-G. Kim, H.-K.
Cho, Microelectron. Eng. 127 (2014) 86.
[15] D. Rio, C. Constancias, M. Saied, B. lcard, L. Pain, J. Vac. Sci. Technol. B 27 (2009)
2512.
[16] H.I. Smith, J. Vac. Sci. Technol. B 4 (1986) 148.
[17] M. Padmanaban, D. Rentikiewicz, C. Hong, D. Lee, D. Rahman, R. Sakamuri, R.
Dammel, J. Photopolym. Sci. Technol. 18 (2005) 451.
[18] P. Kruit, S. Steenbrink, J. Vac. Sci. Technol. B 23 (2005) 3033.
[19] A. Roy, B. Craver, L.E. Ocola, J.C. Wolfe, J. Vac. Sci. Technol. B 29 (2011) 041005.
[20] K. Suzuki, J. Shears, B. Smith, Microlithography: Science and Technology, CRC
Press, 2010.
[21] C. Mack, J. Vac. Sci. Technol. B 27 (2009) 1122.
[22] D. Drouin, A.R. Couture, D. Joly, X. Tastet, V. Aimez, R. Gauvin, Scanning 29
(2007) 92.