Analytical solutions of film planarization during spin coating

Analytical solutions of film planarization for both trench and ridge during spin coating are presented in order to offer a simple equation to designers and researchers for describing the degree of planarization (DOP). We found that in the case of high Ohm(2) (Ohm(2) > 100), DOP is independent of d/h(f) for the isolated trench and ridge. The governing dimensionless parameter Ohm(2) = rho omega(2)omega(3)r(0)/(gamma h(f)) represents the ratio of centrifugal forces to surface tension forces in the coating where rho and gamma are density and surface tension of the fluid, respectively, omega is rotation speed, r(0) is radius of position of the feature on the wafer, d and omega are height and width of the feature, respectively, and h(f) is liquid film thickness. We also found that for high Ohm(2) (Ohm(2) > 100), there is an identical DOP value for trench and ridge. However, for low Ohm(2) (Ohm(2) < 1), the values of DOP for trench and ridge are different even under a fixed set of Ohm(2) and d/h(f). Under low Ohm(2) conditions, the trends of DOP variation for trench and ridge are almost contradictory. Experimental data proposed by Peurrung and Graves in J. Electrochem. Sec. are shown for comparison, and results from the present analytical solutions agree with their data.