Download The effect of photoresist contrast on the exposure profiles obtained

JOURNAL OF APPLIED PHYSICS 103, 083550 共2008兲
The effect of photoresist contrast on the exposure profiles obtained with
evanescent fields of nanoapertures
Eungman Lee and Jae Won Hahna兲
Nano Photonics Laboratory, School of Mechanical Engineering, Yonsei University, 134 Sinchon-dong,
Seodaemun-gu, Seoul 120-749, Republic of Korea
共Received 27 November 2007; accepted 17 February 2008; published online 28 April 2008兲
We propose a simple theoretical model to predict the exposure profiles of a photoresist obtained with
evanescent fields of nanoapertures. Assuming the electric field intensity to be a Gaussian
distribution function with an exponential decay, the top critical dimension and the depth of the
photoresist profile are described with analytic formulas. The profiles are analyzed as a function of
the photoresist contrast and the electric field intensity decay length. © 2008 American Institute of
Physics. 关DOI: 10.1063/1.2907971兴
I. INTRODUCTION
Optical lithography has played a key role in the progress
of the semiconductor industry over the past several decades
because of its high productivity.1 Optical lithography has
progressed by pushing the optical resolution limit, allowing
the production of high-density electronic devices. In principle, however, the optical resolution is limited by the intrinsic properties of light diffraction.
The recent discovery of extraordinary transmission of
light through a perforated metal film2,3 has offered possibilities for optical nanolithography. The electromagnetic field
around a nanoaperture is coupled with the local surface plasmon excited through the hole in the metal film.4–6 The localized field underneath the aperture creates a bright spot
smaller than the diffraction limit. To enhance transmission
through the nanoaperture, researchers have investigated various aperture geometries, such as C-shaped7 and bow tie
apertures.8 High-transmission nanoapertures also have potential applications in nanopatterning9,10 and high-density optical and magneto-optic data storage.11,12
The electric field around the aperture does not propagate,
instead sharply decaying along the direction of the hole in
the metal film.13 Thus, the photoresist profile created by the
localized electric field is quite different than that formed by a
propagating electric field of conventional image. The intensity threshold model is commonly used to predict the profile
of a photoresist exposed by a localized electric field.9,14–16
This model determines the photoresist profile by using isoexposure dose contour corresponding to the threshold dose of
the photoresist. The intensity threshold method, however, is
not able to model the actual response of the photoresist. It
was reported that the properties of photoresist should be additionally included in the intensity threshold method for
modeling the subwavelength scale photoresist profiles in holographic lithography.17 Therefore, in order to accurately predict the nanosize features of photoresist profiles, we need to
take into account the properties of both the photoresist and
the localized electric field.
In the present work, we suggest a simple theoretical
a兲
Electronic mail: [email protected].
0021-8979/2008/103共8兲/083550/4/$23.00
model to describe the profiles of a photoresist exposed by
evanescent fields of nanoapertures. This model improves
upon the threshold intensity method by including the effect
of the contrast of the photoresist. Assuming the localized
intensity distribution to be a Gaussian function with exponential decay, we analyze exposed photoresist profiles as a
function of the photoresist contrast and of the decay length
of the localized field.
II. MODELING OF PHOTORESIST PROFILE
EXPOSURE BY EVANESCENT FIELD
The contrast of the photoresist is a key parameter in
determining the exposure profile of the photoresist. Upon
absorbing light energy, the photoresist undergoes a chemical
change. The sensitivity of the photoresist is represented by
the contrast curve as a function of the logarithmic exposure
dose 共ln E兲 for a normalized photoresist thickness removed.
The value of the contrast ␥ is defined as the linear slope of
the contrast curve, as shown in Fig. 1. The dash-dotted line
in Fig. 1 is a contrast curve of a real positive resist. The
FIG. 1. 共Color online兲 Normalized photoresist thickness as a function of
ln共exposure E兲 for a positive photoresist exposed and developed under a
lithographic process. The dash-dotted line is a real contrast curve of a photoresist. The solid line 共a contrast curve with a slope兲 and the dashed line 共a
step function兲 are the two theoretical contrast curves used for modeling. The
step function curve represents the intensity threshold model.
103, 083550-1
© 2008 American Institute of Physics
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083550-2
J. Appl. Phys. 103, 083550 共2008兲
E. Lee and J. W. Hahn
photoresist does not respond at exposures below the threshold exposure dose Eth, defined as the minimum exposure
required for the photoresist reacting to light. When the exposure dose reaches clearing exposure dose Ec, the photoresist
is completely removed. Figure 1 also depicts two idealized
types of contrast curves used in our models: the dashed line
is a unit step function and the solid line is a straight line with
a slope between Eth and Ec. From the slope of the curve in
Fig. 1, the contrast may be expressed as18
␥=
z/T0
⌬共T/T0兲
=
,
⌬共ln E兲 ln关E共z兲/Eth兴
0 ⬍ z ⬍ T0
and Eth
⬍ E共z兲 ⬍ Ec ,
共1兲
where ⌬ is an abbreviation for “change in,” T0 is the thickness of the photoresist, T is the thickness of the remaining
photoresist, E is the exposure dose, and z is the depth of
photoresist from the photoresist layer. Assuming that contrast
curve shown in Fig. 1 has a constant slope, we have the
second term on the right hand side of Eq. 共1兲.
We arrange Eq. 共1兲 in the parameter z as
冉 冊
z = T0␥ ln
E共z兲
,
Eth
0 ⬍ z ⬍ T0 .
共2兲
We can rewrite the exposure dose E共z兲 which matches
the depth of removed photoresist at z
再
E共z兲 = Eth exp
冎
z ln共Ec/Eth兲
,
T0
Eth ⬍ E共z兲 ⬍ E0 .
共3兲
To predict the exposure profile of the photoresist, we
assume that the impinging light has a Gaussian intensity distribution and propagates into a very thin photoresist layer. In
this way, we can neglect the divergence and interference of
the light.
Our model predicts that, when an evanescent field is
applied to the photoresist, the resulting profile reflects the
intensity distribution of the field penetrating the photoresist.
Figure 2 shows the exposure dose distributions at several
depths in the photoresist. The peak of the exposure dose
distribution decays along the depth of the photoresist z due to
the decay of the localized electric field. The predicted profiles of the photoresist exposed by the evanescent field are
schematically plotted in the lower part of Fig. 2. For the step
contrast curve, as shown in Fig. 2共a兲, the crossing points of
the line of Eth and the exposure dose distribution curves at
different depths are projected into the photoresist. The profile
resulting from our model using the step contrast curve is the
same as the result of the intensity threshold method, which
determines the photoresist profile by mapping to the contour
of Eth.
In the case of the contrast curve with a slope, the photoresist is partially removed in the region of the exposure
dose distribution between Eth and Ec. The exposure profile is
obtained by the following procedure. We calculate the exposure doses E共z1兲, E共z2兲, and E共z3兲 between Eth and Ec using
Eq. 共3兲. As shown in the upper portion of Fig. 2共b兲, we find
the intersection points of the exposure dose line E共z兲 with the
exposure dose distribution curve at the corresponding photoresist depth z. When we project the intersection points into
FIG. 2. 共Color online兲 The intensity distributions of the localized electric
fields at several layer depths z in the photoresist 共upper plots兲 and the photoresist exposure profiles 共lower plots兲. 共a兲 Photoresist profile for a step
contrast curve. 共b兲 Photoresist profile for a contrast curve with a slope.
the photoresist, we obtain intersection points of the projection lines with the photoresist layer at the corresponding
depth z. After then, as shown in the lower part of Fig. 2共b兲,
we obtain the profile of the trench in the photoresist by connecting the intersection points.
To calculate the exposure profile, we consider the intensity distribution of the evanescent field of the nanoaperture.
In general, the transmitted intensity is calculated by the
Bethes– Bouwkamp diffraction theory19,20 for a circular aperture in a thin perfect conductor plate illuminated by an
incident plane wave.
The theoretically predicted transmitted intensity has a
smooth distribution function at near field except at the edge
of the circular aperture and exponentially deceases with increasing distance from the circular aperture.21
However, most of the intensity distributions passed
through the arbitrary shaped nanosize apertures cannot be
described with analytic functions. Therefore, for simplicity
of exposure profile calculations, we assume that the intensity
distribution is described with Gaussian distribution function
in the x-direction and exponentially decreasing along the
z-direction with a decay constant ␤ to describe the nonpropagating property of the evanescent field. Then, we describe
the intensity distribution I 共x, z兲 and the exposure dose distribution E 共x, z兲 in the photoresist with the following equations:
2
I共x,z兲 = I pe−wx e−z/␤ ,
共4兲
2
E共x,z兲 = I共x,z兲t = E pe−wx e−z/␤ ,
共5兲
where I p is the peak intensity at x = z = 0, w is the constant
related to the width of the Gaussian distribution function, t is
the exposure time, and E p = I pt is the peak exposure dose at
x = z = 0. In order to obtain the depth profile of the photore-
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083550-3
J. Appl. Phys. 103, 083550 共2008兲
E. Lee and J. W. Hahn
FIG. 3. 共Color online兲 Profiles of the photoresist exposed by the localized
electric field for the step contrast 共dashed line兲 and the contrast with a slope
共solid line兲. Dash-dotted lines are isoexposure dose contours in the
photoresist.
sist, we substitute Eq. 共5兲 into Eq. 共2兲. Rearranging the equation in terms of spatial parameters x and z, we obtain
z=
冋冉 冊 册
␥T0
Ep
ln
− wx2 = A − Bx2 ,
1 + ␥ T 0/ ␤
Eth
where
A=
冉 冊
␥T0
Ep
ln
1 + ␥ T 0/ ␤
Eth
and B =
共6兲
w␥T0
.
1 + ␥ T 0/ ␤
Using Eq. 共6兲, we can readily derive the top critical dimension 共CD兲 and the depth of the profile. The top CD is the
width of the trench at the photoresist surface 共z = 0兲 given by
top CD = 2x0 = 2
冑 冑 冉 冊
A
=2
B
1
Ep
.
ln
w
Eth
共7兲
The depth of the photoresist profile along the z axis 共x
= 0兲 is obtained as
D=
冉 冊
␥T0
Ep
ln
.
1 + ␥ T 0/ ␤
Eth
共8兲
In Fig. 3, we plot the profiles of the photoresist exposed
by a localized electric field for the two types of contrast
curves. The dashed line represents the photoresist profile for
the step contrast, which matches the profile predicted by the
intensity threshold method. The solid line in Fig. 3 represents
the equation z = A − Bx2 关Eq. 共7兲兴, which is the photoresist
profile for the contrast with a slope. From Eqs. 共7兲 and 共8兲,
we find that the top CD does not depend on the contrast of
the photoresist, but the depth of the profile is a function of ␥.
FIG. 4. Depth of the photoresist profile calculated as a function of the
photoresist contrast ␥ for peak exposure doses E p of 40, 60, 80, and
100 mJ/ cm2.
deviation in the depth of the profile of the photoresist exposed with the localized electric field increases by 60% as ␥
varies from 1 to infinity.
Since a higher E p creates a larger projection area where
the exposure dose exceeds Eth, the top CD and the depth of
the photoresist profile D simultaneously increase along with
E p. The decay length ␤ of the intensity, which decays along
the axis of the aperture, is a fundamental parameter of the
evanescent field. When we estimate the decay length of the
power density transmitted a nano size circular aperture,21 we
obtain the decay length of ⬃55 nm for a circular aperture of
50 nm in diameter. Therefore, we choose the range of the
decay length from 10 to 60 nm to describe the evanescent
field intensities of nanoapertures. According to Eq. 共7兲, the
depth of the photoresist profile D depends on both ␥ and ␤.
Figure 5 shows the depth of the photoresist profile calculated
as a function of the decay length for various contrasts. In the
case of ␥ = ⬁, D is linearly dependent on ␤. In Fig. 4, we find
III. RESULTS OF CALCULATIONS
For the calculation of photoresist profiles, we assume a
Gaussian intensity distribution with a photoresist thickness
T0 of 50 nm, decay length ␤ of 30 nm, Gaussian distribution
constant w of 0.001, and the threshold dose Eth of
20 mJ/ cm2. We calculate the depth of the photoresist profile
D as a function of the contrast ␥ for E p from
40 to 100 mJ/ cm2. The results are plotted in Fig. 4. As expected from the theory, the depth D increases with ␥ and the
depth of the photoresist profile for ␥ = ⬁ matches that obtained by the threshold intensity method. We find that the
FIG. 5. Depth of the photoresist profile calculated as a function of the
intensity decay length for contrasts of ␥ = 1, 3, 7, and ⬁. The peak exposure
dose E p is 80 mJ/ cm2.
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083550-4
J. Appl. Phys. 103, 083550 共2008兲
E. Lee and J. W. Hahn
that the difference between the curves of D for ␥ = 1 and ␥
= ⬁ becomes larger as ␤ increases. In the case of ␤ = 10 nm,
the increase of D is 20% for the variation of ␥ from 1 to ⬁
but 120% in the case of ␤ = 60 nm. Thus, the effect of the
contrast in the modeling of the photoresist profile becomes
more important as ␤ increases.
IV. CONCLUSIONS
In conclusion, we propose a simple theoretical model to
describe the profile of a photoresist exposed to an evanescent
field through the nanoaperture. Including the effect of the
contrast of the photoresist in the model allows us to more
accurately describe the profile. We introduce a schematic diagram to show the difference in photoresist profiles obtained
by our model and by the intensity threshold method. Assuming the evanescent field intensity to be a Gaussian distribution function having an exponential decay, we derive analytic
formulas for the top CD and the depth of the photoresist
profile. It is found that the depth of the photoresist increases
by 60% for the variation of the contrast ␥ from 1 to infinity.
In practice, the error in the depths of the photoresist profiles
predicted using the intensity threshold method is estimated to
be 20% for the photoresist S1805 共␥ = 3兲, popularly used for
recording nanoscale patterns by plasmonic lithography.9,10,22
For the proposed model, we find that the depth of the photoresist profile monotonically increases with the decay length
of the localized intensity distribution, and the effect of the
contrast in modeling the photoresist profile becomes more
important as the decay length increases.
Finally, we note that the proposed model can readily
treat the absorption of light in the photoresist since absorption is generally expressed with the same type of exponential
decay function. Furthermore, we expect that this model can
be easily extended to predict the nanoscale profiles of a photoresist exposed by the evanescent field distribution in an
arbitrary shape by combining the results of numerical analysis, such as the finite differential time domain method.
ACKNOWLEDGMENTS
This work was supported by “Nano R&D Program” of
Korea Science and Engineering Foundation 共Project No.
M10609000019-06M0900-01910兲 and “Development Program of Nano Process Equipments” of Korea Ministry of
Commerce, Industry and Energy 共Project No. 100302592007-01兲.
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