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A Predictive and Corrective Model for Bulk Heating Distortion in Photomasks
(Ref. only 3653 SPIE00 BS ms 000102)
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A Predictive and Corrective Model for
Bulk Heating Distortion in Photomasks
Bassam Shamoun, David Trost, and Frank Chilese
Etec Systems, Inc., 26460 Corporate Avenue, Hayward, CA 94545 USA
Phone: (510) 783-9210; fax: (510) 887-2870; e-mail: bshamoun@etec.com
ABSTRACT
Finite element (FE) numerical models were proposed to simulate and predict substrate thermal expansion in
photomask substrates and were found to be computationally expensive and dependent on the mask-writing
strategy. The present work describes a newly developed model that predicts and corrects for the substrate
heating effects in the photomask. This proposed model provides a practical way of predicting in-plane
distortions during real-time patterning that is not limited to any writing strategy or pattern density
distribution. The main advantage of this model is that it significantly reduces the computational time by
using the linear superposition theory. By adopting the concept of linear superposition, pattern placement
errors of mask substrates can be determined at any time during writing using lookup tables from
precomputed FE models. If the thermal distortion of the substrate at the time of writing is known, beam
deflection can be introduced to correct for the distorted substrate. The results predicted by the linear
superposition FE model showed a difference of less than 10% compared with those predicted using a real-
time calculation FE model, in a worst case scenario. The accuracy of the linear superposition FE model was
found to be partially dependent on the size of the simulated patterning field. The results presented in this
paper illustrate the effect of other parameters on the performance of the newly developed model, such as
the shape of the patterning fields and pattern coverage uniformity. The overview of this work focuses on
fused silica mask substrate materials.
Keywords:
photomask, electron beam, substrate heating, linear superposition, finite element, pattern
placement, thermal load, temperature rise, in-plane distortion, coverage
1. INTRODUCTION
Analytical and numerical models
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were developed in the past for studying the thermal behavior of the
photomask substrate due to bulk heating. Thermal loading-induced distortions during photomask patterning
using electron-beam (e-beam) lithography constrain the desired range of operating conditions of the
pattern-generation tool. To meet the design requirements
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of the pattern generation tool for the 100 nm
node and below, mask substrate heating effects must be minimized. Mask substrate heating, which is
caused by e-beam energy deposition during patterning, results in thermal expansion and pattern placement
errors. This is different from the local resist heating that affects resist sensitivity. Numerical finite element
(FE) models are used to calculate thermal-related photomask distortions due to substrate heating. This
paper discusses a new technique to predict and correct for substrate bulk heating distortion in photomasks.
2. FINITE ELEMENT SIMULATIONS
The substrate heating of a 6-in. photomask, shown in Figure 1, was simulated in the past using FE
analysis.
3,4
In the simulations, the total energy deposited in the mask substrate is determined from e-beam
parameters (absorbed dose and acceleration voltage). The pattern area of the mask is divided into an array
of fields, with each field subdivided into a specified number of finite elements. Thermal load, in the form of
equivalent surface heat flux, is applied to each field in the pattern area to simulate the e-beam energy
deposition in the substrate. The temperature rise of the mask substrate is calculated as a function of time
and used to determine the displacement of each field from its original position prior to patterning. The (Ref. only 3653 SPIE00 BS ms 000102)
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magnitude and the direction of each displacement vector are a function of several parameters, such as the
absorbed dose (percentage coverage), patterning time, thermal and mechanical boundary conditions, and
writing strategy.
By tracking the displacement of each nodal point in the pattern area of the substrate for a given number of
thermal load steps, the in-plane distortion map is obtained. This is a direct calculation method in real time
that requires a relatively large computation time. This makes the implementation of most FE-based models
to predict placement errors during patterning somewhat difficult. To minimize the computation time, efforts
were made to employ faster solvers and use coarse-meshed FE models while maintaining accuracy.
However, different mask-writing strategies and difficulty encountered in determining pattern density prior
to patterning makes implementation of the real-time FE calculation method even harder. It is noteworthy
that the writing strategy varies from one mask to another and is usually determined by customers based on
the pattern data file.
Figure 1. A schematic diagram of a 6-in., SiO
2
photomask. The locations of the kinematic mounts are shown with the
translational degrees of freedom constrained at each mounting point given in parentheses. All dimensions
are in millimeters.
( ,z)
( ,z)
( ,z)
132.0
152.4
6.35
In the present work, we propose a new technique
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to predict and correct for pattern placement errors of
substrate heating that is more practical and easy to implement. This technique is based on the concept of
linear superposition theory, which overcomes all of the previous problems within an acceptable percentage
of error. That is, no prior knowledge of writing strategy or pattern density distribution is required. The main
idea is to estimate the amount of e-beam energy deposited into the mask substrate at any time from pattern
data collected during writing. This data can then be used to manipulate precomputed FE models. An equal
offset in the placement error of a written feature can be introduced for each expected distortion during
patterning so that when the substrate cools (i.e., is no longer distorted), the written feature appears at the
correct position.
Lookup tables are created in which distortion maps for known pattern coverage are calculated. Each
distortion map describes the thermal and mechanical responses of the substrate at a given time step. These
lookup tables are saved in the memory of a computer attached to the operating system of the MEBES
®
mask-writing machine. The distortion data from these tables are used to predict placement errors during
writing by directly extracting and linearly scaling distortion vectors based on units of energy deposited at a
given time step. Because bulk heating is a dynamic process, the deployment of the superposition technique
to predict the distortion requires knowledge of mask patterning. In other words, at any given time step
during writing, the distortion of the entire substrate is obtained by adding the contribution of the newly
patterned fields to the previous data propagated in time. (Ref. only 3653 SPIE00 BS ms 000102)
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As described earlier, the substrate is divided into an array of coarse fields in the FE model. Each field is
approximately the size of a square centimeter. The selection of a field size is dependent on the thermal
diffusion length of the substrate materials and required accuracy of predicted results by the linear
superposition model. A series of FE models is computed in which a unit of energy is introduced into one
field. Accurate distortion maps are then calculated at regular time intervals on the order of one minute. This
process is repeated for each field in the substrate, and the resulting array of distortion maps is stored. The
FE software ANSYS
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is used to perform all calculations. Figure 2 is a schematic diagram that illustrates
how lookup tables for distortion maps are created.
Figure 2. A schematic diagram showing a construction of lookup tables used to correct for substrate placement error
due to heating: (a) a pattern area of the 6-in., SiO
2
mask divided into 10
×
10 coarse fields, and (b) an array
of in-plane distortion maps for selected patterned fields at different time steps. All dimensions are in
millimeters.
Time = 3 t
Time = n t
Time = i t
Time = t
Time = 2t
(a)
(b)
132.0
132.0
IPD map
ith field
exposure
Field
Time = 3 t
Time = n t
Time = i t
Time = t
Time = 2 t
IPD map
jth field
exposure
Time = 3 t
Time = n t
Time = i t
Time = t
Time = 2 t
IPD map
kth field
exposure
2
10
11 12
20
45 44
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The linear superposition technique was proven valid as long as the substrate temperature rise at any time
during patterning is small compared to the surrounding temperature. This is true as, under most stringent
conditions of photomask substrate heating, the maximum temperature rise is only a fraction of a degree.
A series of analyses was performed to prove the concept of linear superposition and determine the accuracy
of the new FE model. The relationship between the in-plane distortion and the pattern coverage of the mask
was determined. The effect of field geometry in the FE simulations was also investigated. Finally, a
comparison between results predicted by real-time calculation FE models and linear superposition FE
models was made. (Ref. only 3653 SPIE00 BS ms 000102)
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2. LINEAR SUPERPOSITION MODELING TECHNIQ