Spring06 web article1

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Lithography

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Predicting Line Edge Roughness through a Mechanistic Model Mark D. Smith, KLA-Tencor Corporation

As feature sizes continue shrinking, yield-limiting phenomena stemming from the molecular nature of photoresist materials—such as line-edge roughness (LER)—have become evident. This article presents a model that explicitly takes into account the molecular nature of the photoresist during the exposure and post-exposure bake (PEB) processes.

A master equation was written that first describes the probability that acid molecules are generated during exposure, and then describes the evolution of the acid, quencher, and blocking-group probability distributions during the bake process. The article also shows how all of the parameters in this model can be derived from the parameters in a calibrated PROLITH continuum model, and used for prediction of LER.

Introduction

LER has received much attention in the literature. Many correlations between process parameters and LER have been investigated experimentally, and two observations seem to shed the most light on the origins of LER. First, the amount of LER is inversely proportional to the square root of the exposure dose. Statistical mechanics tells us that fluctuations are proportional to the square root of the average number of particles. This type of dependence on the dose implies that shot noise leads to a random distribution of acid during the exposure process, and that this propagates through the resist processing to ultimately create LER. The second important observation is that LER is strongly correlated with metrics of image quality, such as aerial image contrast or NILS. For example, both Michaelson1,2 and Nikolsky3 published experimental results for LER that correlated LER with the predicted concentration gradients at the end of PEB. Nikolsky used the concentration gradients predicted by PROLITH simulations, while Michaelson used an aerial image model combined with a model for exposure

and the PEB process. This approach allowed Michaelson to reduce a very large set of LER experiments onto a single curve. The picture of LER that arises from these observations is that while shot noise leads to a random dose distribution, the consequences of this randomness can be mitigated by high contrast images, which are less sensitive to random fluctuations.

Michaelson’s papers (and others) also attempt to characterize how the formulation of the photoresist impacts LER. The most dramatic trend is that LER decreases with quencher loading, with LER being especially excessive for resist formulations with no quencher.

One possible explanation for this is that increased quencher loading increases the required dose, which would generate a relatively lower amount of fluctuations due to shot noise. However, Michaelson found that LER correlated only with the concentration gradients at the end of PEB – all of the different quencher loadings fell on the same LER versus gradient curve, without an explicit dependence on dose.

Several theoretical models for LER have also been proposed. The paper by Gallatin4 is especially outstanding because it explains many of the observed trends. One shortcoming of the model is that it did not include quencher loading. This is the main outstanding issue addressed in this investigation.

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Predictions of the continuum models in PROLITH

The approach in this study is to combine a deterministic (continuum) calculation of the blocked polymer gradient with a probabilistic model for the consequences of shot noise. Both models will be based on the exposure and PEB models in PROLITH. PROLITH uses the Dill exposure model: (1) In the Dill exposure model, PAG is the relative concentration of photoacid generator, c is the Dill photospeed parameter, I is the intensity in the resist, and E is the exposure dose. The model in PROLITH for PEB is a reaction-diffusion system, and is believed to be an accurate continuum model of the PEB process:

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paper, as well as additional data collected during the course of the LER study5. In his paper, an ESCAP-type resin was combined with 2.8 wt% TPS-nonaflate as a photoacid generator, and various amounts of triethanol amine as a quencher. Focus-exposure matrices were collected for 160-nm lines on a 320-nm pitch on a Canon ES4 stepper, with a numerical aperture (NA) of 0.8 and a partial coherence of 0.8. Dose-to-size and LER data were presented for this formulation as well, with the LER data collected at best focus and the dose-to-size for each resist formulation. These values, along with a comparison with the PROLITH predictions, are reproduced in Table 1. Michaelson’s analysis of the LER data started with evaluating the continuum kinetic equations (2) in the absence of acid or quencher diffusion. The PROLITH continuum model for PEB does not have this constraint, so we can evaluate the impact of diffusion on the blocked polymer gradients at the end of PEB. As shown in Figure 1, the blocked polymer gradient increases as the quencher loading is increased.

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(2)

In this continuum model, B is the blocking group relative concentration, H is the acid relative concentration, and Q is the quencher relative concentration.

Blocked Polymer Fraction

0.9

No Quencher 5% Quencher 10% Quencher 20% Quencher

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

We have carefully calibrated a PROLITH resist model (which also includes a develop model) to the resist formulation presented in the paper by Michaelson on LER with a deep ultraviolet (DUV) photoresist. The calibration data consists of the data presented in the

Quencher Loading

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Figure 1: Fraction of blocked polymer at the end of PEB for several quencher loadings as predicted by the calibrated PROLITH model. Each image corresponds to best focus and dose-to-size for the different resist formulations.

PROLITH Dose-to-Size (mJ/cm2) Experimental LER (3s, nm)

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Table 1: LER and dose-to-size data for selected resist formulations in the Michaelson study.

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Review of the Gallatin Model for LER

Here we briefly review the LER model proposed by Gallatin4. This model is built on the concept of the “resist blur”6, which is basically the response of the PEB model to a point source of acid. This “point source” could be thought of as a single acid molecule that diffuses randomly though the resist, performing deblocking reactions at the same time. The predicted concentration of acid is then:

(3)

In this equation, r is a position vector, r0 is the position of the point source, and N is the dimension (e.g., 1D, 2D, or 3D). This equation is only valid if the motion of the acid molecules is random and other constituents of the photoresist do not interfere with the motion of each acid, i.e. the acid molecules do not interfere with each other and no quencher is present. If we have only a single point source, we can substitute this Green’s function into the equation for the deblocking reaction and obtain the equation for the blocked polymer fraction:

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The probability of finding a deblocked site at a particular point in the resist is the product of the probability of having an acid molecule from exposure and the resist blur function, summed over all possible acid locations in the resist. A linearized form of the critical ionization model is used to predict how these probability distributions impact develop.

In essence, the Gallatin model investigates the consequences of random events during the exposure process (shot noise) by using a continuum description of PEB and develop processes. However, the model does not account for the presence of quencher, because the resist blur function is only valid when nothing interferes with the acid. The model described in the current study does not have these limitations. Here, the starting point is the same probabilistic description of the exposure process; then, a probabilistic description of the PEB process is applied. This probabilistic PEB model describes the interactions between the acid, quencher, and blocking groups in detail. However, the current model is limited in that is does not explicitly account for acid diffusion and quencher diffusion. These effects can easily be added to the model at a later time.

0.25

(4)

The relationship between the above equation and the resist blur is that the resist blur is the deblocked fraction, or 1-B. The resist blur function originally published by Hinsberg6 was for 1D diffusion, and an example of this function is shown in Figure 2.

The LER model by Gallatin starts with an exposure model for the probability that an acid molecule is generated at a particular point in the resist. This model gives a probability distribution of acid molecules that follow Poisson statistics. The average number of acid molecules in the distribution is given by the solution to the Dill exposure model, equation (1). The rest of the model uses a deterministic representation of the PEB and develop processes. To evaluate the impact of PEB on the initial random distribution of acid molecules, the resist blur function is evaluated for each acid location.

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Figure 2: Resist blur function for 1D diffusion with an acid diffusion coefficient (DH) of 36.2nm2/sec, an amplification rate constant (kamp) of 0.316 1/sec, and a PEB bake time of 60 seconds. We obtained these parameters by calibration of a PROLITH resist model to the data in the LER study by Michaelson1.

The master equation

The blocked polymer fraction results in the previous section clearly demonstrate that the “chemical contrast” increases with quencher loading, and this is consistent with the idea that the large gradients in the blocked polymer fraction decrease LER. In this section, a probabilistic model is proposed that is based on a set Spring 2006

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of master equations7 for the probability that a specific number of acid, quencher, and blocked polymer molecules will be found within a cubic volume of resist. The master equation for the PEB model given by equation (2) is given by:

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(9)

(5) P is the probability that there are nH acid molecules, nQ quencher molecules, and nB blocked polymer sites.

The functions t+ and t - represent transition probabilities. t+ is the probability that some other state will be converted into the state with (nH, nQ, nB ) molecules.

We will assume that there are two ways this can happen. First, we could have a deblocking reaction that is catalyzed by the presence of acid. This reaction would decrease the number of blocked sites by one, but leave the number of acid and quencher molecules unchanged: (6) The probability that this occurs is proportional to the product of the amplification rate constant, the number of acid molecules present (nH ), the number of blocking sites present, and the probability of having the state (nH, nQ, nB +1): (7) Note that ka is different from kamp in equation (2) by a conversion from normalized acid and blocking fractions to a rate constant appropriate to deal with numbers of molecules in the control volume. A similar transition can occur due to the quenching reaction, from a state (nH+1, nQ+1, nB ) to the current state. Combining these two possible transitions gives the equation for t+:

(8)

After writing a similar equation for t - (nH, nQ, nB ), a master equation can be written for state (nH, nQ, nB ) : 4

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Notice that the evolution equation for P(nH, nQ, nB ) depends on the probabilities that other states exist.

For this reason, an equation must be written for these states as well, and this leads to a very large set of coupled master equations that must be solved simultaneously.

The initial condition for the above set of master equations is constructed by calculating the average number density of blocking sites, the average number density of quencher molecules, and the average number density of PAG molecules from the resist formulation. The number density of acid molecules is calculated by solving the Dill exposure model, equation (1), using the dose given in Table 1 and an intensity of 0.2622 – the intensity at the line edge calculated in PROLITH. These number densities are converted to an average number of molecules by selecting a control volume.

The control volume can be considered to be the “pixel size” of the calculation – for large control volumes, we would expect to get results that are the same as the continuum model for PEB, but as the control volume becomes smaller, the molecular nature of the resist will become more apparent. We arbitrarily choose our control volume as a cube that is 10 nm on a side.

Once we have the number of each type of molecule expected to be in our control volume, we can construct an initial probability distribution by assuming Poisson statistics.

The above master equation is nonlinear, so it is solved numerically. Example results are shown in Figure 3 for the resist formulation with no quencher present. For such a low exposure dose, the average number of acid molecules in a 153nm3 volume of resist is 2.2 molecules.

This means that there is a significant probability (10.6 percent) that no acid molecules will be present inside the volume of resist. The peak on the far right centered at about 2500 molecules is the probability that no deblocking occurs – for all of the resist formulations, the average number of blocking sites is 2555 in a 153nm3 volume of resist. The next peak over corresponds to the probability that one acid molecule was present, so about one quarter of the sites were deblocked.

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The next peak, centered around 1600 blocked sites, corresponds to the presence of two acid molecules, etc.

It is interesting to contrast this result with the results we obtain with a larger control volume of resist. Shown in Figure 4 are the blocked polymer fraction probabilities for a control volume that is 10 nm on a side and for volumes 20 nm and 30 nm on a side. (Note that the probabilities have been normalized by the volume size so that all three probabilities can be easily viewed on a single graph.) For the 203nm3 volume, the average number of acid molecules is 5.3, and for the 303nm3 volume, the average number is 17.9 molecules. The difference between the three probability distributions is dramatic: for the larger volumes, we now have several closely spaced peaks instead of a few peaks that are further

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apart. The trends shown in this figure demonstrate the key contributor of the LER – the quantized nature of the exposure process and the resist formulation itself.

It is easy to see from these graphs that LER was likely present at previous technology nodes, but the dimensions of interest in this study (the “pixel size”) were large enough that the discrete nature of the exposure process and of the resist materials was not important.

Next, this study returned to the original control volume size of 153nm3 of resist, examining the impact of quencher loading. Results are shown for all four of the resist formulations in Figure 5 for the original control volume size. Here, the “chunkiness” of the distribution is reduced, and when quencher is present, each of the discrete peaks broadens and forms a long tail on the left side. This broadening of the peaks should correspond to a reduction in LER. The first peak, which corresponds to a low level of deblocking, grows in size and becomes dominant for the larger quencher loadings. However, the broad tail to the left of the main peak remains. The range of number of blocked sites corresponding to the tail overlaps the portion of the develop rate curve where the resist switches from insoluble to soluble. Shown in Figure 6 are the distributions in Figure 5, along with the develop rate curve used in the PROLITH model.

From the figure, it is apparent that there is always a probability that a resist volume will switch from insoluble to soluble, but the part of the tail in the switching portion of the develop rate curve is reduced as quencher is added.

Figure 3: Probability of the occurrence of a specific number of blocked sites inside a 153nm3 volume of resist at the end of PEB for the resist formulation with no quencher.

Probability of Blocked Site 0.003 No Quencher 5% Quencher 10% Quencher 20% Quencher

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Figure 5: Probability of the occurrence of a specific number of blocked sites at the end of PEB. The initial distribution of acid, quencher, and blocked sites was

Figure 4: Probability density for the occurrence of a fraction of the number of

determined by selecting the exposure dose listed in Table 1 for each resist

blocked sites inside a 10 nm cube, a 20 nm cube, and a 30 nm cube. The

formulation.

probabilities have been normalized by the control volume size so that they can all be easily viewed on a single graph.

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This result is very similar to the equation derived by Gallatin4 and by Byers8. By estimating the magnitude of the perturbation as the standard deviation of the distributions shown in Figure 4 with the blocked polymer gradients at the line edge shown in Figure 1, the correlation shown in Figure 6 can be obtained.

Although the constant of proportionality is obviously not incorrect, the correlation is very good. When spatial variations and diffusion are included, more quantitative results should be obtained directly from the solution to the master equation.

Number of Blocked Sites 18

Figure 6: The probability distributions shown in Figure 5 along with the

In order to calculate a final estimate for the LER, this model must be extended to include the spatial variations throughout the photoresist, not just the distribution at a single point near the line edge. This will correspond to solving the master equations on a lattice of volumes throughout the resist with the inclusion of a pair of new terms in the master equation that account for spatial diffusion of the acid and quencher molecules. Because this will require a much larger state-space (the populations at all of the grid points), it will be addressed in future work.

In the meantime, a simple estimate for LER can be determined with the results in this study. This can be done by considering a Taylor series expansion of the blocked polymer fraction at the line edge: (10) We can re-write this equation for a perturbation to the blocked polymer fraction, and re-arrange to find the new line edge position:

(11)

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Experimental LER (3 Ďƒ, nm)

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corresponding develop rate model for this photoresist.

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Figure 7: Estimated LER from the standard deviation of the distributions in Figure 5 divided by the gradient at the line edge in Figure 1.

Conclusions and future work

This article presented calculations with PROLITH and calculations with the master equation for the PEB process to demonstrate how resist formulation impacts LER. The PROLITH calculations, based on a continuum model, show that increasing the quencher loading causes an increase in the blocked polymer gradient at the line edge. This continuum result supports the conclusion by Michaelson that LER is mitigated by high contrast images in the resist at the end of PEB. Next, the master equation was solved for a small volume of resist at the line edge, and these results showed the discrete nature of the exposure process and the acid-catalyzed deblocking reaction during PEB. The addition of quencher broadened the sharp peaks in the blocked polymer distribution, which will lead to a decrease in LER.

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Future work will connect these two models by solving the master equation on a lattice of volumes throughout the entire line-space pattern. This will require a much larger state-space (the populations at all of the grid points) and the addition of diffusion to the master equation.

This will give a complete picture of LER, and allow greater insight into the impact of imaging conditions, process conditions, and resist formulation on LER. References 1. T.B. Michaelson, A.T. Jamieson, A.R. Pawloski, J. Byers, C.G. Willson, “Understanding the role of base quenchers in photoresists” Proc. SPIE, 5376 (2004).

2. T.B. Michaelson, A.R. Pawloski, A. Acheta, Y. Nishimura, C.G. Willson, “The effects of chemical gradients and photoresist composition on lithographically generated line edge roughness” Proc. SPIE, 5753, 368-379 (2005).

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4. G. Gallatin, “Resist blur and line edge roughness” Proc.

SPIE, 5754, 38-52 (2005) 5. Private communication.

6. W.D. Hinsberg, F.A. Houle, M.I. Sanchez, J.A. Hoffnagle, G.M. Wallraff, D.R. Medeiros, G.M. Gallatin, J.L. Cobb, “Extendibility of chemically amplified resists: another brick wall?”, Proc. SPIE, 5039, 1-14 (2003) 7. C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, 6th Edition, Springer 2002 8. J. Byers, C.A. Mack, M.D. Smith, “Defining process windows for line-edge roughness and pattern collapse” SPIE 2005 (unpublished)

3. P. Nikolsky, R. Tweg, E. Altshuler, E.N. Shauly, “LER characterization and impact on 0.13-mm lithography for OPC modeling” Proc. SPIE, 1323-1331 (2004)

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