Mathematical model of spin-coated photoresist on a spherical substrate

Xiao-guo Feng; Lian-chun Sun

doi:10.1364/OPEX.13.007070

1. Introduction

There are many applications for concave gratings and diffractive optical elements (DOE), including light splitting, imaging, and focusing [1]. These optical elements are usually fabricated by the photolithography technology [2]. Thus, the problem of coating uniform photoresist film on a curved substrate is brought forward. However, few papers analyze it in detail until now [3]. Its main reason should be the complexity of curved equation derivation and the particularity of application.

The paper focuses on the evolution of film thickness of spin-coated photoresist on a spherical substrate and shows the derivation course in detail. Experiment results can affirm the accuracy of these equations and the method. Thus, these equations may be used for predicting the film thickness of spin-coated photoresist on a spherical substrate, and the derivation method can also be applied for other curved surfaces. Furthermore, the theoretical results can also guide the development of new equipment.

2. Flow mechanism

The flow mechanism of spin-coated photoresist on a spherical substrate relates to the centrifugal force, the coriolis force, the viscous force, the surface tension, the volatilization of solvent, and so on. Thus, following assumptions are required

Dilute photoresist is Newtonian fluid approximately
A spherical surface is an axial symmetry shape, so the coriolis force may be ignored [4]
Because the film thickness is quite small in comparison with the characteristic dimension of substrate, the surface tension may also be ignored
The shear force that arose from wind is also ignored.
The motion equation of spin-coated photoresist on a plane is given by [5]
$- μ \frac{\partial^{2} u}{\partial z^{2}} = ρ ω^{2} r$
In Eq. (1), ρω ² r is the centrifugal force,μ is the kinematic viscosity of fluid, u is the radial velocity in z direction, z is the height in the film thickness direction, ρ is the density of fluid, ω is the angular velocity, and r is the radial position.
Obviously, on a spherical substrate, there is (see Fig. 1)
$\sin θ = \frac{r}{R}$

Fig. 1. The force diagram of infinitesimal fluid
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In Eq. (2), θ is the pitch angular of spherical tangent line in a radial position (obviously, θ ≺ 90°),R is the spherical radius. Considering the effect of gravity, we may derive the motion equation of spin-coated photoresist on a spherical substrate
$- μ \frac{\partial^{2} u}{\partial z^{2}} = ρ ω^{2} r \cos θ - g \sin θ$
In Eq. (3), g is the gravity force of infinitesimal fluid, and it relates to the density of fluid. Because the lubrication approximation of fluid is satisfied, the following boundary conditions can be gained.
On a substrate (z equal to zero), there is
$u (r, z, t) = 0 .$
The velocity gradient of a free surface is zero, and there is
$\frac{\partial u}{\partial z} (r, z, t) = 0$
Thus, the integral expression of Eq. (3) can be derived
$u = \frac{r}{μR} (- \frac{1}{2} z^{2} + hz) (ρ ω^{2} \sqrt{R^{2} - r^{2}} - g)$
The mass continuity equation of incompressible fluid is
$\frac{\partial h}{\partial t} + \frac{1}{r} \frac{\partial}{\partial r} (r \int_{0}^{h} udz) = 0$
According to Eq. (6) and Eq. (7), there is
$(2 g - 2 ρ ω^{2} \sqrt{R^{2} - r^{2}} + \frac{ρ ω^{2} r^{2}}{\sqrt{R^{2} - r^{2}}}) \frac{h^{3}}{3 μR} = \frac{\partial h}{\partial t} + \frac{r h^{2} (ρ ω^{2} \sqrt{R^{2} - r^{2}} - g)}{μR} \frac{\partial h}{\partial r}$
The differential expression of the optional position's film thickness is
$\frac{dh}{dt} = \frac{\partial h}{\partial t} + \frac{\partial h}{\partial r} \frac{dr}{dt}$
Compare Eq. (8) with Eq. (9), two evolution equations of photoresist thickness and radial position can be gained
$\frac{dh}{dt} = (2 g - 2 ρ ω^{2} \sqrt{R^{2} - r^{2}} + \frac{ρ ω^{2} r^{2}}{\sqrt{R^{2} - r^{2}}}) \frac{h^{3}}{3 μR}$
$\frac{dr}{dt} = \frac{r h^{2}}{μR} ρ ω^{2} \sqrt{R^{2} - r^{2}} - g)$
Eq. (10) indicates that the change speed rate of film thickness disaccords among radial positions, and this is also different of the sphere’ and the planar’. As the radial position is determined, the integral expression of Eq. (10) is
$h = h_{0} {[1 + (2 ρ ω^{2} \sqrt{R^{2} - r^{2}} - 2 g - \frac{ρ ω^{2} r^{2}}{\sqrt{R^{2} - r^{2}}}) ∙ \frac{2 h_{0}^{2} t}{3 μR}]}^{- \frac{1}{2}}$
In Eq. (12), h ₀ is the initial film thickness of a radial position. Eq. (12) describes the variety that the initial film thickness (h ₀) relates to the radial position (r).
According to Eq. (3), flow-out necessary condition of fluid film is
$ρ ω^{2} r \cos θ \geq g \sin θ$
The solution of Eq. (13) is
$r \leq \sqrt{R^{2} - \frac{g^{2}}{ρ^{2} ω^{4}}}$
While r = 0 , the film thickness can’t be calculated in Eq. (12) (see Eq. (7)). Actually, the film thickness of revolving spindle is non-uniform forever [6].
So, there is
$0 < r \leq \sqrt{R^{2} - \frac{g^{2}}{ρ^{2} ω^{4}}}$
According to Eq. (12), while r ~ 0.816R, there is
$h = h_{0} {(1 - \frac{4 g h_{0}^{2} t}{3 μR})}^{- \frac{1}{2}}$
Provided that $\sqrt{R^{2} - g^{2} / ρ^{2} ω^{4}}$ ≻ 0.816 R , the position of r ≈ 0.816R is a key point according to Eq. (12). While 0 ≺ r ≺ 0.816R , the fluid film thins gradually by the centrifugal force, and h ≺ h ₀. However, while 0.816 R ≺ r ≺ $\sqrt{R^{2} - g^{2} / ρ^{2} ω^{4}}$ , the fluid film thickens gradually by the gravity force, and h ≻ h ₀. Thus, the spin-coated form should not be used provided that the caliber of a spherical substrate is larger than 1.632R (see Fig. 1), or else the uniform film can’t be gained forever.
While R ≻≻ r or high speed spin-coated, the effect of the gravity force may be ignored, the variety that the film thickness can be gained according to Eq. (12)
$h = h_{0} {[1 + \frac{4 ρ ω^{2} h_{0}^{2} t}{3 μ}]}^{- \frac{1}{2}}$
Obviously, Eq. (17) is the case of spin-coated fluid film on a plane (See Ref. [5]).
In order to gain the solid film, the volatilization of solvent must be considered. So more assumptions are necessary, including
The initial concentration of solvent is uniform, and the concentration of solute (c) is changed uniformly by the volatilization of solvent, and c is independent of r
The change of c in the direction of z may be ignored
The density of solvent and solute is uniform, and the capacity of solution equals to the sum of the solvent capacity (L) and the solute capacity (S).
The actual fluid disaccords with these assumptions, but the difference has no effect on the results [7].
According to these assumptions, there is
$c (t) = \frac{S}{(S + L)}$
$h = S + L$
The change of viscosity arises from the change of concentration, and it has an effect on the dynamics characteristic of fluid. Following equations are the change rate of S and L owing to the flow-out of fluid and volatilization of solvent
$\frac{dS}{dt} = - c (2 ρ ω^{2} \sqrt{R^{2} - r^{2}} - 2 g - \frac{ρ ω^{2} r^{2}}{\sqrt{R^{2} - r^{2}}}) ∙ \frac{h^{3}}{3 μR}$
$\frac{dL}{dt} = - (1 - c) (2 ρ ω^{2} \sqrt{R^{2} - r^{2}} - 2 g - \frac{ρ ω^{2} r^{2}}{\sqrt{R^{2} - r^{2}}}) ∙ \frac{h^{3}}{3 μR} - e$
$h_{f} = S_{f}$
In Eq. (21), e is the speed rate of solvent volatilization, and h_f is final film thickness, and S_f is final solute thickness, obviously, h_f = S_f.
With the volatilization of solvent, the concentration of fluid increases. The viscosity of fluid increases, also. So the rheological change of fluid should be expressed by the function of power-law
$μ = μ_{solvent} + μ_{solids} c^{γ}$
The volatilization of solvent may be ignored while the film thickness is thinner than the critical point h _1/3 (it relates to the initial condition). While the film thickness is h _1/3, the volatilization of solvent and the radial flow have same effect on the variety of film thickness [8]. Moreover, e relates to the angular velocity of spin-coated (See Ref. [6]).
$e = C \sqrt{ω}$
In Eq. (24), ω is the angular velocity of spin-coated, and C is the coefficient of laboratory and device.
Thus, the final film thickness may be attained
$h_{f} = S_{f} = c_{0} h_{\frac{1}{3}} = c_{0} {[\frac{3 μR ∙ C \sqrt{ω}}{(1 - c_{0}) (2 ρ ω^{2} \sqrt{R^{2} - r^{2}} - \frac{ρ ω^{2} r^{2}}{\sqrt{R^{2} - r^{2}}} - 2 g)}]}^{\frac{1}{3}}$
In Eq. (25), c ₀ is the density of nonvolatile matter in the photoresist.
Commonly, as the photoresist is spun coating on a spherical substrate, the assumption of Newtonian fluids and laminar airflow (i.e., some assumptions of solvent volatilization) can be satisfied, the results of Eq. (25) are reasonable. As the spin-coated speed or the caliber of substrate is quite big, Sukanek' work must be considered seriously [9].

3. Experiment and results

The positive photoresist BP212-6 is diluted with the thinner, and is spun coating on a concave spherical substrate. Afterwards, the photoresist is exposed by a laser direct writing’ equipment, and a generating line figure is generated by concentric optical scan. After developing, in some radial positions, the film thickness is tested by an atomic force microscope (See Fig. 2). Experiment shows that the mathematical model is accurate.

4. Conclusions

Based on some assumptions, we present the evolution equation of spin-coated photoresist on a spherical substrate. Considering the effect of solvent volatilization, we derive the final film thickness of spin-coated photoresist on a spherical substrate. Moreover, two inferences may be gained

In Eq. (25), the final photoresist thickness is independent of initial film thickness, so it is independent of the initial material amounts. While the radius of a spherical substrate and the angular speed are confirmed, the final film thickness relates only to the radial position, and the film thickness gets thick along with the increased radial dimension. Thus, the form of spin-coated must be altered to gain the even film thickness while r/R ≻ 0.816. For example, the opening of spherical substrate turns towards flank when the photoresist is spun coating.
Obviously, these conclusions may be also applied to other similar fluid.

Fig. 2. Final photoresist thickness on a concave spherical substrate with a spherical radius of 20mm, the angular velocity of 2000rpm. C _o =0.015 , g ≈ 0 , μ/ρ ∙ C √ω = ve = 0.03 m ²/s ².

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Acknowledgments

This study is supported by the National Defence Science Advanced Foundation No.10.4.2.ZK1001).

References

1. J. H. Burge, D. S. Anderson, T. D. Milster, and C. L. Vernold, “Measurement of a convex secondary mirrorusing a holographic test plat,” in Proc. SPIE 2199, 193-198 (1994). [CrossRef]

2. Yongjun XIE, Zhenwu LU, Fengyou LI, Jingli ZHAO, and Zhicheng WENG, “Lithographic fabrication of large diffractive optical elements on a concave lens surface,” Opt. Express 10, 1043-1047 (2002). [PubMed]

3. S.B.G. O’Brien and L.W. Schwartz, “Theory and modeling of thin film flows,” Encyclopedia of Surface and Colloid Science, Marcel Dekker, New York , 5283-5297 (2002).

4. T G Myers and J P F Charpin., “The effect of the coriolis force on axisymmetric rotating thin film flows,” Int.J. Non-linear Mech. 36, 629-635 (2001). [CrossRef]

5. A. G. Emslie, F. T. Bonner, and L.G. Peck, “Flow of a viscous liquid on a rotating disk,” J. Appl. Phys. 29, 858-862 (1958). [CrossRef]

6. A Acrivos, M J Shah, and E E. Petersen, “On the flow of a non-newtonian liquid on a rotating disk,” J. Appl. Phys. 31, 63-968 (1960). [CrossRef]

7. D Meyerhofer, “Characteristics of resist films produced by spinning,” J. Appl. Phys. 49, 3993-3997 (1978). [CrossRef]

8. YUE Hongda, PAN Longfa, and BIN Yuejing, et al, “Mechanics analysis in CD-R dye coating process,” in Proc. SPIE 4930, 253-257 (2002). [CrossRef]

9. Peter C. Sukanek, “Spin Coating,” J. Imaging Technol. 11, 184-190(1985).

Mathematical model of spin-coated photoresist on a spherical substrate

Abstract

1. Introduction

2. Flow mechanism

3. Experiment and results

4. Conclusions

Acknowledgments

References

Cited By

Figures (2)

Equations (25)

Optics Express