Determining film stress from substrate shape using finite element procedures

A method for determining stress in a film applied to a top surface of a substrate, comprising:

• (a) dividing the substrate into substrate plate elements having plate element nodes set at the top surface of the substrate;
• (b) dividing the film into film plane elements having plane element nodes set at an interface with the substrate plate element nodes;
• (c) determining displacements of the substrate plate element nodes, denoted by a displacement matrix {Ds};
• (d) determining forces at the substrate plate element nodes, denoted by a matrix {Fs}, from the displacement matrix {Ds} and a substrate stiffness matrix [Ks];
• (e) determining forces acting on the film at the film plane element nodes, denoted by the matrix {Ff}, from the determined forces at the substrate plate element nodes {Fs} as {Ff}=−{Fs};
• (f) determining film displacements of the film plane element nodes, denoted by the matrix {Df}, from the determined forces acting on the film at the film plane element nodes {Ff} and a film stiffness matrix [Kf];
• and
• (g) determining the film stress for the film plane elements from the determined film displacements.

A system for determining stress in a film applied to a top surface of a substrate, comprising:

• (a) a measuring device adapted to measure at least one of an out-of-plane displacement, cross-section rotation, or curvature of the substrate at a plurality of points on the substrate;
• (b) a computer coupled to the measuring device and adapted to receive measurements therefrom and programmed to:
• (i) determine a shape of the substrate from the at least one measured out-of-plane displacement, cross-section rotation, or curvature;
• (ii) determine forces at the top surface of the substrate from the determined shape of the substrate;
• (iii) determine forces acting on the film from the determined forces at the top surface of the substrate;
• (iv) determine film displacements from the determined forces acting on the film;
• and
• (v) determine the film stress from the determined film displacements.