Calculational study on membrane mask distortion and correction

We have developed an analytical technique that predicts both in-plane and o ut-of-plane distortion arising from an arbitrary stress distribution on a m embrane surface. Our technique can also solve the inverse problem; i.e., we can predict the stress distribution which, when applied to any existing di stortion, eliminates it. These techniques, based on the variational method, require one to formulate the total energy that results from the membrane d istortion, which is a straightforward task even for complicated stress dist ributions. One then iteratively searches for the distortion map that minimi zes this total energy, which, according to minimum energy principle, should be the true membrane distortion for a given stress distribution. The inver se problem is solved analytically by performing a functional minimization o n the total energy and solving for the stress distribution. This calculatio nal technique should enable us to correct for arbitrary distortions by, for example, applying another film or differential heating. (C) 2001 American Vacuum Society.