Electron beam heating of a thin film on a highly conducting substrate

An analysis is made of the steady-state temperature distribution in a poorly conducting plane film on a highly conducting semi-infinite substrate, owing to a time-independent heat input in a cylindrical region of the film and substrate. The problem is of interest in connection with the localized hardening of anodic oxide films on silicon by electron beam bombardment in order to produce oxide diffusion masks for the manufacture of integrated circuits. A formal solution is obtained for arbitrary dependence of the heat input on radius and depth, and a detailed study is made of a particular case in which the heat input is independent of radius across the beam, and varies in a realistic manner with depth in the film. Approximate formulas are given for the temperature in the film when the radius of the beam is large compared to the thickness of the film, and also when the conductivity of the film is small compared to the conductivity of the substrate. The approximate formulas are compared with the results of calculations based on the exact solution. Finally, a crude estimate is made of the time required to reach the steady state.