Thin film approximation model speeds up resist simulation
02/01/2003
By: Robert Wildfeuer, Juriy Malov, Henning Müllerke, Thomas Schmöller, Christian K. Kalus, SIGMA-C GmbH, Munich, Germany
Today's optical lithography simulators can typically calculate areas of about 10 ¥ 10µm2. Progress in resist simulation has focused on modeling new chemically amplified resist systems and not on speeding up the algorithms [1, 2].
The situation is quite different in image formation; several authors have come up with fast aerial image calculations, increasing the speed by several orders of magnitude with only a small loss in accuracy [3, 4]. Fast resist images have not been available, though, most likely because a stepper/scanner is an optical projection system describable by exact mathematics; the same is not true for a resist.
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Resist behavior depends on the reaction kinetics and diffusion processes during post-exposure bake (PEB) and development. All resist models available today use a set of more or less well-understood chemical parameters.
There is a growing need to develop an algorithm generating resist images with high precision at considerably higher speed. In contrast to lumped parameter models, an alternative algorithm must be designed to properly predict 3-D scenarios, where the resist surface propagates simultaneously in all directions according to a speed function that depends on the contrast, the (x, y) distribution of deposited energy, and where the z-dependence is given by the thin film approximation (assuming a homogeneous chemical composition) for which Lambert's law holds [5, 6].
Model basics
The model used — called fast resist image-thin film approximation (FRIT) — is based on Lambert's law, which states that the image intensity decreases exponentially with the depth due to absorption within the resist. This z-dependence governs the speed with which the resist surface propagates in the vertical direction. FRIT is a compound model for vertical light propagation, PEB, and resist
development, whereas in a full simulation, these steps are modeled separately. Either case requires the aerial image as input.
FRIT can be understood by its similarity to the vertical propagation model (VPM) in a currently available lithography simulator, SOLID-C. Both models start from only one aerial image at the top of the resist. Therefore, the image transfer to the resist is symmetrical with respect to the defocus. FRIT will not only be compared to VPM but also to the scaled defocus model (SCDF), taking the SCDF simulation as the reference point. Accuracy of the FRIT model in relation to the "real world" is the main interest. (Whenever the abbreviations VPM or SCDF are used below, they represent a shortened form of a full simulation sequence.)
 Benchmark on "big" areas up to 32 ¥ 32µm2. Resolution in x, y is 21.25nm. |
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Calibration of the model
Conditions throughout the test using the UV6 resist are shown in the table.
The physical model contains the coefficients of threshold dose Dth, dose D, an effective absorption coefficient aeff, effective resist thickness deff, and a sensitivity g. These coefficients are easily accessible, but since the model has been derived from the thin film approximation, it would be expected that they have to be fitted for use in a bulk resist.
A significant result of the evaluation is that only one coefficient, the sensitivity g, drifts away from its expected value. The calibration, therefore, consists of no more than fitting the sensitivity by using an experimental contrast curve or, as has been done here, by using full simulation.
Accuracy
All tests performed use as a reference a full simulation with SOLID-C starting with the aerial image calculation, followed by a PEB and resist development utilizing the Weiss model. The calculations with FRIT use the same aerial image used for the full simulation. Due to the absence of experimental data, the results of the SCDF model with the finest resolution of 5nm are used as "reality." The test scenario consisted of isolated lines, dense lines, contact holes for 250 and 150nm technology nodes, and various defects.
In order to save CPU time, it was necessary to find the largest grid possible that still provides the needed resolution. It appears that FRIT converges faster than the other models. At around 25nm (wafer scale), it has reached its asymptotic end value and no longer changes. In contrast, the SCDF and the VPM models show a convergence at about 10nm.
The comparison of process windows shows that FRIT is in relatively good agreement with VPM. This indicates at least that FRIT is an alternative to vertical propagation, the fastest among the conventional models.
More interesting, however, is a comparison with "reality" as given by the 5nm SCDF results: The agreement of all models in focus is very good. Deviations beyond 10% confidence appear only for small contact holes, but these deviations can also be found in the VPM results. Out-of-focus, the agreement between VPM and FRIT is still good, but both curves deviate from reality. FRIT is a thin film approximation, rendering best results in the limit of a thin resist.
Time performance
All benchmarking was done with the following system specification: Windows 2000 O/S, 756MB main memory, and 1.1GHz CPU. In this section, the development step includes all relevant calculations to get the resist profile if the aerial image has already been calculated.
In detail, the development step in full simulation includes a) exposure of the resist including reflection and standing wave calculations, b) all PEB calculations, especially acid diffusion, and c) resist formation using the recalculated inhibitor distribution. In contrast, FRIT calculates the resist formation directly from the given aerial image at the top of the resist without notice of any concentration behavior inside the resist. The time consumption of the development step between the FRIT model and the full simulation is compared below. Because there is almost no difference in time consumption between the scaled defocus and vertical propagation models, only the scaled defocus data is shown.
The figure on p. 40 compares the CPU time of the development step between full simulation and FRIT for a resolution of 21.25nm. In general, time consumption is linear with simulation area for both full and FRIT simulations since the slope in the double plot is about one. (If the memory consumption gets larger than the main memory of the computer, the increase in CPU time will no longer be linear, however.) FRIT has a speed advantage over SCDF by two orders of magnitude.
Conclusion
The FRIT model, using a thin film approximation, makes very good predictions in focus — as good as those obtained with a full simulator. Deviations from the real value are significantly smaller than 10%; it has limits to predict small contact holes within 10% accuracy. The reference used is SOLID-C with a scaled defocus model. Throughout focus, FRIT is at least as good as the so-called vertical light propagation in combination with a full resist model. Values for out-of-focus CDs typically deviate from real values by more than 10%. This issue has been addressed by the SLAB algorithm, which introduces defocus effects. FRIT has a speed advantage over conventional full resist simulation by two orders of magnitude.
References
- S. Robertson, et al., "An Improved Notch Model for Resist Dissolution in Lithography Simulation," SPIE, 4345, pp. 912–920, 2001.
- H. Fukuda, K. Hattori, T. Hagiwara, "Impact of Acid/Quencher Behavior on Lithographic Performance," SPIE, 4346, pp. 319–330, 2001.
- D. Cole, et al., "Derivation and Simulation of Higher Numerical Aperture Scalar Aerial Images," Jap. J. Appl. Phys., 31, 4110, 1992.
- Y.C. Pati, A.A. Ghazanfarian, R.F. Pease, "Exploiting Structure in Fast Aerial Image Computation for IC Patterns," Proc. BACUS/SPIE, 1995.
- C. Mack, "Enhanced Lumped Parameter Model for Photolithography," SPIE, 2197, 501, 1994.
- T.A. Brunner, R.A. Ferguson, "Approx. Models for Resist Processing Effects," SPIE, 2726, 198, 1996.
Thomas Schmöller can be reached at SIGMA-C GmbH, Thomas-Dehler-Str. 9, 81737 Munich, Germany, ph 49 089 630 257 40, fax 49 089 630 25749, e-mail [email protected].