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Measurement and modeling of solvent removal for spin coating.

INTRODUCTION

Spin coating a thin polymer photoresist film is employed as a part of the lithographic process in the microelectronics industry. Often a planar coating is required on a substrate with topography. Currently, post-spin processing, e.g., etch-back and polishing, must be employed to achieve the desired planarization.

The coating planarization may be improved by decreasing the rate at which the solvent leaves the film. During spinning there is simultaneous solution spinoff and solvent evaporation. The coating viscosity increases exponentially as the solvent evaporates. As a result, the coating becomes so viscous that the capillary force, a planarization force, becomes negligible. If solvent could be retained within the coating for longer time, coating planarization might be improved.

To quantify the effect of solvent concentration on planarization, it is important to know the change of solvent concentration during spinning. There have been continuous efforts in modeling of spin-coating process in the past three decades. Emslie et al. (1) modeled the spinning a Newtonian nonvolatile fluid. Sukanek (2) modeled spin coating a volatile solution arid considered a uniform coating concentration, which implied no diffusion resistance within the coating. Bornside (3) further included coating diffusivity in his model. However, each of these models was evaluated only on the basis of comparison of the predictions with dry coating thicknesses, which is of course only at the boundary conditions. A more rigorous comparison for any model is to compare the model predictions with experimental results over a more complete time frame. This was the goal of this research.

Spin-coating experiments were performed using solutions of a high-molecular-weight ([M.sub.w] = 310,000) polystyrene (PS) in toluene. While the use of PS, especially of such high [M.sub.w] is unlikely for actual spin-coating applications, this system is useful in understanding further the fundamentals of the spin-coating process because such an understanding requires data for the variation of physical properties (e.g., viscosity, vapor pressure, diffusivity) with solvent concentration. "Real" resist formulation and data on the variation of physical properties with composition are not generally available. Furthermore, prediction of these properties is complicated by the fact that they are often composed of mixed solvents. The PS-toluene system is much better characterized and was, therefore, chosen as a model system since data exist for the composition dependence of physical properties. Since we had previously (4) measured the variation of viscosity and glass-transition temperature [T.sub.g] with composition for the PS ([M.sub.w] = 310,000)-toluene, we could use this system in our simulation. The benefit of this approach is that results can be directly linked to fundamental physical properties, which can be used in resist formulation, rather than being limited to a specific resist.

An experimental measurement technique for evaluating coating solvent concentration during spinning is of practical importance. in this work, an on-line optical technique is used to measure the solvent concentration and film thickness during spinning. A gel permeation chromatography (GPC) off-line technique, which provides an accurate measurement, is performed to evaluate the accuracy of the optical measurement.

The spin-coating model is compared with the measured concentration and thickness during spinning as well as dry coating thickness. Modeling spin coating involves solving a moving boundary problem in partial differential equations. However, under certain conditions, the problem can be significantly simplified. These conditions are discussed.

MODELING

The spin-coating process is shown schematically in Fig. 1. The process is a nonsteady-state, coupled momentum, mass and energy transport process. Because a complete solution is complicated, several simplifying assumptions are made in the following development. Photographs taken by Daughton and Givens (5) showed that most of the solution initially deposited on the wafer was sloughed-off in the first few rotations, leaving only a thin coating for the rest of the process. Inertial force is thus negligible compared with the viscous force because the coating is thin. Since the coating thickness was much smaller than the substrate radius, the lubrication approximation could be adopted. Since the coating flow is dominated by centrifugal force, the Coriolis force and gravity potential are negligible. The coating density is assumed to be a constant. The velocity equations obtained from the simplified momentum equations are then:

[Delta]/[Delta]z([Mu] [Delta][u.sub.r]/[Delta]z) = -[Rho][[Omega].sup.2]r (1)

1/r [Delta]/[Delta]r(r[u.sub.r]) + [Delta][u.sub.z]/[Delta]z = 0 (2)

[u.sub.r[where]z=0] = 0 [u.sub.z[where]z=0] = 0 [Delta][u.sub.r]/[Delta]z[where]z = h = 0. (3)

No slip at the substrate and inviscid air are expressed in the boundary conditions. Because the coating is thin, coating heat transfer is low near steady state; thus, the process can usually be considered as isothermal and no energy equation is necessary (2, 3). The solvent concentration in the coating is a function of the z coordinate alone under two conditions: if there is uniform evaporation above the coating surface and if the coating thickness is uniform with radial position. The first condition requires laminar air flow above rotating disk, which is satisfied for small-diameter substrates. The second condition requires a Newtonian viscosity. As discussed later, this is often a good approximation for most of the process. The one-dimensional mass conservation equations are

[Delta][[Rho].sub.s]/[Delta]t = [Delta]/[Delta]z (D [Delta][[Rho].sub.s]/[Delta]z) - [u.sub.z] [Delta][[Rho].sub.s]/[Delta]z (4)

[Delta][[Rho].sub.s]/[Delta]z[where]z=0 = 0 -D/1 - [x.sub.s] d[[Rho].sub.s]/dz[where]z=h = [k.sub.s][p.sub.i] (5)

[p.sub.i] = [p.sub.0][[Phi].sub.s](1 - [[Phi].sub.s] + [Chi][(1 - [[Phi].sub.s]).sup.2]. (6)

The boundary condition at z = 0 dictates that there is no mass transfer through the wafer while the boundary condition at z = h is the equation for the overall mass flux at the free surface for a frame of reference that translates with the same velocity as the free surface (6). The mass transfer coefficient [k.sub.s] suggested by Sparrow and Gregg is used (7). The expression for the solvent vapor pressure is derived from the Flory-Huggins model. The coating thickness is given by:

[Delta]h/[Delta]t = [u.sub.z[where]z = h] - [k.sub.s][p.sub.i]/[Rho]. (7)

The above model is similar to Bornside's one-dimensional model (3). If the diffusion coefficient is large enough, mass transport is evaporation-controlled and the model then becomes Sukanek's model (2). If there is no solvent evaporation, the model is equivalent to the model of Emslie et al. (1).

Figure 2 shows the theological data (4) for PS)([M.sub.w] = 310,000)-toluene solutions which were used in the model by first correlating the data using the Carreau model (additional low polymer concentration viscosity data used in the model are not shown in [ILLUSTRATION FOR FIGURE 2 OMITTED]). Figure 2 shows the fit obtained using the concentration-dependent Carreau model parameters. The mutual diffusion coefficient data are from two sources. The first was PS ([M.sub.w] = 270,000)-toluene self-diffusion coefficient data from Pickup and Blum (8) at low polymer concentration, which were converted to mutual diffusion coefficient using the method of Duda et al. (9). The other source was PS ([M.sub.w] = 210,000)-benzene sorption data at high polymer concentration from Odani et al. (10), which were converted to mutual diffusion coefficient using the procedure described by Crank (11). Data for benzene, which has a similar molecular size as that of toluene, were used as an approximation because high polymer concentration diffusion coefficient data were not available for PS-toluene over a wide concentration range. At sufficiently high polymer concentrations the mutual diffusion coefficient is not sensitive to molecular weight (12) so literature data are used without correction. Figure 3 shows the data and the fit of the data using a polynomial correlation.

EXPERIMENTAL

Spin Coating

Wafers of 125 mm (5 inch) were coated with PS-toluene solutions. The PS was reported by the manufacturer as having a [M.sub.w] of 310,000 and [M.sub.w]/[M.sub.n] of 3.1. A Solid State Model 140 spin coater, set at full acceleration and full brake, was used. The spin speed was calibrated by an optical tachometer. Solutions were deposited while the wafer was at rest and then the wafer was spun for 30 s. The coatings were baked at 110 [degrees] C for 60 min to remove the remaining solvent. Dry coating thickness was measured using a Dektak IIA (Sloak Corp.) profilometer with a 12.5-[[micro]meter] diameter stylus. The thickness obtained in these experiments was uniform in the radial direction, and the film surface was smooth and without any noticeable defects (e.g., striations, pinholes).

Concentration and Thickness Measurements

Optical Method

Laser light is highly coherent, so it does not have thickness measurement limitation. The optical method is used to measure the coating refractive index. The laser interferometer arrangement is illustrated in Fig. 1. The incident angle is small, so normal incidence can be approximated. A 1 mW He-Ne (Uniphase) unpolarized laser, 623.8-nm wavelength, was used as the light source. The reflected light intensity was measured by a silicon photodiode detector PIN-10P (United Detector Technologies). A Tektronix 2211 oscilloscope was used to record the detector output. The change of output is only 10 mV, compared with the detector output range, 0-1 V. A linear response is expected within this small voltage range.

Figure 4 is the optical interferogram obtained for spin coating a 10% solution at 1000 rpm. As the film thickness decreases during spinning, the interference intensity oscillates sinusoidally with extrema at maxima R1(t), and minima R2(t). The film thickness change between a given maximum (or minimum) and the previous maximum (or minimum) is given by

[Delta]h = [Lambda]/2[(n[(t).sup.2] - [sin.sup.2][Phi]).sup.1/2]. (8)

The disadvantage of this laser interferometer, compared with a spectrometer, is that it can only measure thickness change between two peaks, not absolute coating thickness. Absolute thickness is determined by adding Ah, from Eq 8, to the thickness at the end of spinning, which contains 20% solvent as measured by GPC. This amount is in agreement with our previous spin-coating experiments (13), which showed that, based on the change in weight during baking, there was up to 20% solvent retained in the coating after spinning. This would be expected since, for this system, solutions become glassy at room temperature at around 20% solvent (4), at which point the diffusion coefficient drops by several order of magnitudes thus dramatically reducing the rate of loss of solvent from the film.

The amplitudes of the extrema R1(t) and R2(t) also change with time, which can be related to the coating refractive index of the film. An expression for the coating refractive index can be derived from the envelope method by neglecting the absorption index (14). The envelope method, where the film thickness was a constant and the frequency was a variable, was originally developed for spectrometer measurement. When applying the envelope method to the laser interferometer, the thickness is time dependent and the wavelength is fixed. Two envelopes, represented by [R.sub.1] and [R.sub.2], are then drawn to connect maximum and minimum points of the sinusoid waveforms. The coating refractive index at a given time can then be determined by

[Mathematical Expression Omitted]

The coating concentration can then be calculated from n(t) by assuming that the n(t) is given by the linear summation of the product of the mass fraction and the refractive index of polymer. Refractive index is found to be a good approximation of concentration for many polymer solutions at low polymer concentration (15).

The accuracy of the optical measurement is affected by three major factors. One is rotation noise. A bare uncoated wafer will produce a background noise during spinning, and this noise will be superimposed on the envelopes. This background noise is shown on the center of Fig. 4. The second factor is the diffusion-caused concentration gradient within the coating. The magnitude of interference waveforms for the film with the same average concentration but different concentration gradient are different. Without the knowledge of the concentration profile, error will be introduced by assuming an uniform coating concentration. It can be shown that the concentration gradient has no effect on calculating coating thickness using Eq 8. The third factor is whether the boundary is optically sharp as required by Eq 9. During spinning, there is spin-induced air flow and rapid solvent evaporation above coating surface. Therefore the air-coating boundary may not be optically sharp. This factor is difficult to be analytically quantified. The details of the technique and application of this optical method are described in another paper (16).

GPC Method

The solvent concentration in the film at a given spin time is determined by spinning for a given amount of time and then quickly transferring the coated wafer to a container of tetrahydrafuran (THF). The coating solvent is thus captured within the THF. After the coating is dissolved, the solution is analyzed using a Waters GPC- 150C to measure the ratio of PS to toluene in the solution. The concentration at different times is determined by repeating these experiments for several spin times. A special light-weight wafer holder was built to reduce the time required for the wafer to stop spinning. As a result, the amount of solvent evaporated between the cessation of spinning and the time when the coated wafer is transferred to the THF is negligible.

The refractive index detector that was supplied with the GPC was not sensitive enough in the low concentration range, so a Waters 484 UV detector, which has a very stable baseline, was used instead. GPC calibration curves for PS and toluene are shown in Figs. 5 and 6, respectively. Because the data form smooth curves and the measurement deviations are very small, the GPC measurements are found to be accurate for determining the absolute concentration of polystyrene and toluene in these solutions.

RESULTS AND DISCUSSION

Concentration Measurements

Figure 4 shows the optical interferogram obtained for coating a 10% solution at 1000 rpm. The concentrations determined from Fig. 4 using Eq 9 are shown in Fig. 7. The GPC-measured concentration, also shown in Fig. 7, is used for comparison.

The results show that the optical method indicates the inflection point in the concentration curve measured with the GPC method. The GPC curve can be approximated by two line segments. The first line, from 0 to 10 s, has a small slope. The second line, starting from about 30% PS, has a larger slope. Finally the slope then decreases, at about 80% PS, as the material becomes glassy. The laser measurement responds to the large slope line, but is not sensitive to the small slope line associated with higher evaporation rate. The small concentration change information is lost in the optical measurement. It can be shown that the errors caused by rotating noise and concentration gradient are no larger than 10% in our case. Thus, it is concluded that optically nonsharp air-coating boundary is the major cause of error.

Concentration Predictions

Figures 8 and 9 compare concentrations predicted by the model with those experimentally measured using the GPC method for 8 and 10% solutions respectively. Curve "a" in the Figures uses the regressed literature diffusion coefficient data of Pickup and Blum (8) and Odani et al. (10) in Fig. 3. The model prediction agrees well with the experimental data before the PS concentration reaches 70%. Above 70% PS concentration, however, the model predictions are well below those measured by GPC.

The discrepancy between model and experiment arises from the modeling of the later stages of the process. This is illustrated by separating the thickness change caused by solution spin-off from that caused by solvent evaporation. Figure 10 illustrates each contribution for coating the 8% solution as determined by the model. Spin-off is found to cease after 4 s spinning. Thereafter, the thickness change is only caused by evaporation.

Much closer agreement between the simulation and GPC results was obtained by arbitrarily modifying the diffusion coefficient correlation away from that following Odani's data in the high concentration range. Figure 3 shows the modified curve. As shown by curve "b" in Figs. 6 and 7, the model agrees well with the measured data using this modified diffusion coefficient.

While the modification to the diffusion data was made arbitrarily in order that the model fit the GPC data better, this modified coefficient may actually be a better approximation to the "real" mutual diffusion coefficient data for PS-toluene than Odani's data. This is demonstrated by first observing (Figs. 5, 8 and 9) that the discrepancy starts from about 70% concentration, which is close to the 80% PS concentration reported (4) as the composition for glass transition at room temperature for PS ([M.sub.w] = 310,000)-toluene. Thus, the discrepancy occurs when the film is below [T.sub.g]. As noted earlier, Odani's data are for PS-benzene and were used because there are little data for the diffusion coefficient of PS-toluene below [T.sub.g]. Frick et al. (17) have measured and recently reported the self-diffusion coefficient of tracer dyes in PS-toluene solutions which can be correlated with toluene self-diffusion coefficients. They report data for three compositions at the temperature of the present experiments (T = 298K) which have been converted to mutual diffusion coefficient data as described earlier and shown in Fig. 3. In addition, Vrentas and Vrentas (18) have recently extended the free-volume theory for prediction of self-diffusion coefficients to glassy systems. Using the same values for the parameters given in that reference and a value of the parameter (not specified in the reference) of A = 536.5K, obtained by a linear regression of [T.sub.g] vs. composition data given in paper by Frick et al. (17), and converted to mutual diffusion coefficient as before yields the diffusion coefficient curve shown in Fig. 3. The modified diffusion coefficient and predictions from the free-volume are in close agreement except from about 0.3 to 0.6 solvent fraction. Over this range, using the parameter developed, the theory tends to underpredict the experimental data.

Figure 11 shows the prediction of spinning a 10% solution at different spin speed. The effect of spin speed on coating concentration is observed. Coating thickness is reduced as spin speed increases, which also contributes to the rapid change of concentration.

Film Thickness Prediction

As average coating polymer concentration approaches 80% polymer concentration, the coating becomes glassy, at which point mass transfer becomes diffusion-controlled. The model predicts the formation of a thin "skin" of nearly dry polymer on the coating surface. After that, the thickness changes very slowly so computation is then stopped. Dry coating thickness is determined by multiplying the wet coating thickness by the average polymer concentration.

The model predictions of both film thickness vs. time and final dry film thickness are in close agreement with experiments. Figures 12 and 13 are the measured and predicted thickness during spinning for 8 and 10% solutions respectively. For 8 and 10% solutions, the measured dry coating thicknesses are 1.20 [[micro]meter] and 1.80 [[micro]meter], while the predicted dry thicknesses are 1.31 [[micro]meter] and 2.06 [[micro]meter]. This corresponds to errors of 9.2% and 12.6% respectively.

The model predicts a uniform coating thickness in the radial direction, which is supported by our experiments. This seems contradictory to the results of Acrivos et al. (19). For a shear thinning fluid, Acrivos predicted that the thickness decreases from the wafer center to edge, However, Fig. 10 shows that spin-off ceases below 20% polymer concentration where the solution still behaves as a Newtonian fluid. As illustrated in Fig. 2, the solutions were Newtonian for shear rates up to 1 [s.sup.-1] and polymer weight fractions up to 55%. While the magnitude of the shear rate for spin coating 125-mm (5-inch) wafers can be on the order of 1000 [s.sup.-1] for the first 0.1 s, our simulation shows that it becomes less than 1 [s.sup.-1] for the remainder of the process. This explains the observed and predicted uniform coating thickness for this polymer solvent system. In addition, equally good agreement with experiment was obtained by using a Newtonian viscosity model in the simulation as was obtained using a Carreau model.

It can be argued that in certain cases for the prediction of film thickness, diffusivity within the coating can be neglected. The loss of solvent from the film by diffusion can be estimated by (11):

[M.sub.t]/[M.sub.[infinity]] = 2 [(Dt/[Pi][L.sup.2]).sup.1/2]. (10)

Solvent evaporation rate can be estimated by setting t as 1. For spinning a 8% solution at 1000 rpm, L is taken at the end of spin-off, as 12 [[micro]meter]. Equation 10 applies only for a constant diffusion coefficient, which is often a good approximation for low concentration polymer solution. D is taken as 5 x [10.sup.-7][cm.sup.2]/s. The calculated diffusion rate is then 7.4 x [10.sup.4] g/[cm.sup.2]s. Evaporation rate on a rotating disk was given by Sparrow and Gregg (7) as:

m = 4.4 x [10.sup.-6] [MW.sub.s][p.sub.i][[Omega].sup.1/2]. (11)

For toluene spun at 1000 rpm, the evaporation rate is 1.4 X [10.sup.-4] g/[cm.sup.2]s, which is much smaller than the diffusion rate. Obviously the mass transport is evaporation control and concentration gradient is negligible at the end of spin-off. This has been demonstrated in the model including coating diffusivity. This estimation explains why early model properly predicted dry coating thickness, even though the diffusion within the coating is not considered (2). Diffusion becomes important if very high spin speed and high vapor pressure solvent are used. This suggests that for purpose of prediction dry coating thickness, fine uniform coating concentration can be adopted if spin speed and vapor pressure are not very high, which are often satisfied for many applications.

CONCLUSIONS

Coating concentration change during spinning is measured by both an optical method and a GPC method. While the optical measurement indicates the inflection point in the concentration-spin time curve, it is not sensitive to small rates of concentration changes. From the comparison with experimental results for coating concentration, thickness during spinning, and dry coating thickness, it is concluded that the current spin-coating model properly describes the process. Up to the solvent concentration where the film becomes glassy, mutual diffusion coefficient data calculated from literature data produce simulation results that are in close agreement with experimental results. For compositions where the film is glassy, there is little diffusion coefficient data. However, the diffusion coefficient used in the model that gives the best fit to the experimental concentration data is in close agreement with both the literature experimental diffusion coefficient data and the predictions of the free-volume theory. A uniform coating thickness is predicted and is explained by the fluid's Newtonian behavior at low polymer concentration. Spin-coating modeling can be significantly simplified by:

1. using only Newtonian low polymer concentration viscosity data, and

2. assuming a coating of uniform concentration if spin speed and solvent vapor pressure are not very high.

ACKNOWLEDGMENTS

The authors thank IBM Corp. for financial support. The authors are very grateful for this support.

NOMENCLATURE

D = mutual diffusion coefficient.

h = coating thickness.

[k.sub.s] = mass transfer coefficient.

L = coating thickness.

[M.sub.[infinity]][M.sub.t] = initial solvent concentration and accumulated evaporated solvent.

[MW.sub.s] = solvent molecular weight.

[n.sub.s], n = refractive index of substrate, coating.

[p.sub.i] = vapor pressure.

[R.sub.1], [R.sub.2] = waveform envelope.

t = time.

u = velocity.

r, z = coordinates in radial and vertical directions.

[x.sub.s] = solvent mass fraction.

[[Rho].sub.s], [Rho] = solvent concentration, coating density.

[Omega] = spin speed.

[Mu] = shear viscosity.

[[Phi].sub.s] = solvent volume fraction.

[Chi] = polymer-solvent interaction coefficient.

[Lambda] = wavelength.

[Phi] = incident angle.

REFERENCES

1. A. G. Emslie, F.T. Bonner, and L.G. Peck, J. Appl. Phys., 29, 858 (1958).

2. P. C. Sukanek, J. Imaging Technol., 11, 184 (1985).

3. D. E. Bornside, PhD thesis, University of Minnesota, Minneapolis (1988).

4. M. D. Bullwinkel, MS, Clarkson University, Potsdam, New York (1991).

5. W. J. Daughton and F. L. Givens, J. Electrochem. Soc., 129, 173 (1982).

6. D.E. Bornside, C.W. Macosko, and L.E. Scriven, J. Imaging Technol., 13, 122 (1987).

7. E. M. Sparrow and J. L. Gregg, J. Heat Trans., 82, 294 (1960).

8. S. Pickup and F. D. Blum, Macromolecules, 22, 3961 (1989).

9. J. D. Duda, J. S. Vrentas, S. T. Ju, and H. T. Lui, AIChE J., 28, 279 (1982).

10. H. Odani, S. Kida, M. Kurata, and M. Tamura, Rep. Preg. Polym. Phys. Japan, 34, 571 (1961).

11. J. Crank, in The Mathematics of Diffusion, p. 242, Clarendon Press, Oxford, England (1967).

12. J. S. Vrentas and C. H. Chu, J. Polym. Sci., Polym. Phys. Ed., 27, 465 (1989).

13. M. D. Bullwinkel, J. Gu, and G. A. Campbell, J. Electrochem. Soc., in press.

14. D. B. Kushev, N. N. Zheleva, Y. Demakopoulo. and D. Siapkas, Infrared Phys., 26, 385 (1986).

15. M. B. Huglin, J. Appl. Polym. Sci. 9, 3963 (1965).

16. J. Gu, G. A. Campbell and M. D. Bullwinkel, J. Appl. Polym. Sci., in press.

17. T. S. Frick, W. J. Huang, M. Tirrell, and T. P. Lodge, J. Polym. Sci., Polym. Phys. Ed., 28, 2629 (1990).

18. J. S. Vrentas and C. M. Vrentas, Macromolecules, 27, 5570 (1994).

19. A. Acrivos, M. J. Shah, and E. E. Petersen, J. Appl. Phys., 31, 963 (1960).
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Author:Gu, J.; Bullwinkel, M.D.; Campbell, G.A.
Publication:Polymer Engineering and Science
Date:Apr 15, 1996
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