Finite element calculations of strains in passivated metal lines

Passivated metal lines, commonly used in integrated circuits, show thermally induced strains and stresses due to the mismatch of the thermal expansion coefficients of the lines and their surroundings. These strains depend on the geometry and the elastic constants of both, line and surroundings (i.e. substrate and passivation) and - for the case of an array of parallel lines - on the repetition distance of the lines. For a given set of these parameters the strains and stresses can be obtained from finite element calculations using a model with plane strain conditions. In order to obtain the dependences of the strains from these parameters systematic finite element calculations have been made for an infinite array of parallel lines with variations of the geometries and the elastic constants. Since only the volume mean values of the metal strains can be measured easily (X-ray, wafer curvature method (metal stress)), the dependences of these mean values on the different parameters are presented.