Simulations of Dislocation Dynamics in Aluminum Interconnects
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Simulations of Dislocation Dynamics in Aluminum Interconnects Lucia Nicola1 , Erik Van der Giessen1 and Alan Needleman2 1 The Netherlands Institute for Metals Research/Dept. of Applied Physics, University of Groningen, Nyenborgh 4, 9747 AG Groningen, The Netherlands 2 Division of Engineering, Brown University, Providence, RI 02912, USA ABSTRACT A discrete dislocation simulation of plastic deformation in metallic interconnects caused by thermal stress is carried out. The calculations are carried out using a two dimensional plane strain formulation with only edge dislocations. A boundary value problem is formulated and solved for the evolution of the thermal stress field and the evolution of the dislocation structure in the cross-section of the line as cooling proceeds. For lines with a small cross section (height or width less than 1 µm), the local concentration of stresses due to dislocation patterning strongly affects the overall stress build up and relaxation. The results show a clear dependence of the transverse stress development on the line aspect ratio. INTRODUCTION The stress that develops in aluminum interconnects due to thermal mismatch between line, substrate and passivation layer strongly affects interconnect reliability [1]. Theoretical studies have considered the purely thermo-elastic build up of stress [2, 3] as well as the relaxation of stress by plastic deformation [4, 5] using a phenomenological continuum plastic constitutive relation. However, when the line width or height is comparable with the length scale of the dislocation patterns that form, the discrete nature of dislocations cannot be ignored for an accurate prediction of the stress distribution. Therefore, to obtain insight into the relaxation process under such circumstances, we use discrete dislocation plasticity to study the build up and relaxation of stress in metallic interconnect lines under thermal loading. A two-dimensional study is carried out, considering the cross-section of a passivated, single crystal aluminum line embedded in an infinitely long array. The dislocations in the aluminum interconnect are treated as line singularities in an isotropic linear elastic medium, and a set of constitutive rules is incorporated for the glide of dislocations as well as their generation, annihilation and pinning at point obstacles. The solution for the state of stress and deformation is at every increment given as a superposition of two contributions: the known, analytical solution for individual dislocations in infinite space and a non-singular linear elastic finite element solution that enforces the proper boundary conditions. Some preliminary results are presented for various aspect ratios of the line, for a line height of 0.4 µm. These results demonstrate the dependence of the effectiveness of stress relaxation by dislocation motion on the aspect ratio as well as on the size of the line. MODEL We consider an infinitely long array of single-crystalline lines, each having thickness h and width w, perfectly bonded to an infinitely large sub
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