Analysis of average thermal stresses in passivated metal interconnects

Abstract

Volume-averaged thermal stresses in passivated metal interconnects on Si substrates are derived for situations where the thickness to width ratio of the interconnect lines is ''small'' or ''large.'' The analysis provides different components of volume-averaged stresses for the most general case of thermal and elastic anisotropy in the passivation layer, the interconnect line, and the substrate. It is shown that the theoretical predictions, particularly those for the hydrostatic stresses, are in agreement with detailed finite element calculations for a wide range of line and passivation geometries of practical interest. The theoretical predictions of average hydrostatic stresses are also found to be in reasonable agreement with available experimental results for thermal stresses derived from x-ray diffraction measurements on passivated Cu lines. The present theoretical results are shown to be far more accurate than prior stress analyses for periodic passivated lines based on Eshelby's theory of inclusions.

Non-uniform, axisymmetric misfit strain: in thin films bonded on plate substrates/substrate systems: the relation between non-uniform film stresses and system curvatures

“…A biaxial form of Stoney, appropriate for anisotropic film stresses, including different stress values at two different directions and non-zero, in-plane shear stresses, was derived by relaxing the assumption of curvature equi-biaxiality [2]. Related analyses treating discontinuous films in the form of bare periodic lines [4] or composite films with periodic line structures (e.g., bare or encapsulated periodic lines) have also been derived [5][6][7]. These latter analyses have also removed the assumption of equi-biaxiality and have allowed the existence of three independent curvature and stress components in the form of two, non-equal, direct components and one shear or twist component.…”

Some Practical Issues of Curvature and Thermal Stress in Realistic Multilevel Metal Interconnect Structures

“…1b). 17 Since the lines are very wide across the line direction, and the line height is very small compared to both its length and width, the stress state in the lines is basically that of equibiaxial plane stress. Therefore, the volume-averaged stress values along the line (x-direction), and across the line (y-direction) are the same, as are the curvatures in the x-and y-directions, for this limiting line geometry.…”

“…In contrast, analytical predictions provide very good insight into the parameter controlling the deformation process, and also provide reasonably accurate results in a time-efficient manner. [15][16][17] Using EshelbyÕs inclusion theory, Niwa et al 15 modeled the line as a cylinder of elliptical cross-section embedded in an infinite isotropic matrix having the same elastic properties as the line. This model was generalized by Korhonen et al 16 by using different elastic properties between the line and the matrix while still neglecting the substrate.…”

“…These early analyses, based on EshelbyÕs theory, can incorporate aspect ratios of the lines, but fail to capture the effects of the substrate and of the neighboring lines, while completely neglecting sharp edges/corners of the lines. Wikström et al 17 were the first to propose a fully fledged thermoelastic analysis of periodic, alternating, metal/dielectric lines on a substrate. This model considered the two limiting cases, low and high line aspect ratios, which is similar to the analysis employed in the present work.…”

The effect of thin film/substrate radii on the Stoney formula for thin film/substrate subjected to nonuniform axisymmetric misfit strain and temperature

“…A biaxial form of Stoney formula (with different direct stress values and nonzero in-plane shear stress) was derived by relaxing the assumption (v) of curvature equi-biaxiality [Freund and Suresh 2004]. Related analyses treating discontinuous films in the form of bare periodic lines [Wikstrom et al 1999b] or composite films with periodic line structures (for example, bare or encapsulated periodic lines) have also been derived [Shen et al 1996;Wikstrom et al 1999a;Park and Suresh 2000]. These latter analyses have removed the assumptions (iv) and (v) of equi-biaxiality and have allowed the existence of three independent curvature and stress components in the form of two, nonequal, direct components and one shear or twist component.…”