ABSTRACT
With the trench full of DI water, a chemical reagent is dispensed onto the substrate. In the interest of uniformity of processing along the full depth of the trench, it is important that the spatial concentration of the reagent along the trench depth be kept as uniform as possible. For analysis purposes, one may assume that the trench is full of water and the top of the trench is covered with liquid reagent. The reagent may penetrate the trench through either (a) convective or (b) diffusive mixing. In order for convective mixing to take place, the flow characteristics must allow for vortices to exist. The possibility of vortex existence may be ruled out by comparing the relevant geometric parameters to the smallest vortex diameter predicted by Kholmogorov scales (see [2]) using the following equation:
where rj is the smallest length scale that can sustain turbulence (i.e. the smallest vortex diameter possible), / i s a characteristic length of the system (in this case it is the width of a trench), and u and v are the flow velocity and kinematic viscosity, respectively. With relatively large trench width and velocity values of / = 0.5 |nm and u = 1 m/s one obtains r¡/l ~ 1, and with more realistic values of / = 0.25 |jm and u = 0.01 m/s one obtains rj/l ~ 100. Since the smallest possible vortex diameter is noticeably larger than the trench width, it is clear that turbulent mixing cannot take place in the trench. This implies that the reagent is transferred into the trench by diffusion only. The same argument applies to transfer of reagents out of the trench when DI water is dispensed onto the surface in order to stop the reaction.
