Theoretical treatment of void nucleation due to coupling of mechanics and vacancy composition

We present a coupled continuum formulation of mechanics and composition applicable to polycrystalline microelectronic and semiconductor structures. The formulation is thermodynamically-based and accounts for atomic transport, creation and annihilation of species, and the interactions of these processes with local stress and strain. Coupled constitutive field equations are obtained for composition and mechanics by standard thermodynamic arguments. They are incorporated in balance laws, and various boundary value problems for metal self-diffusion and dopant diffusion in silicon can be solved [1,2]. Electromigration is also treated. More recently, we have tried to address the question of void nucleation. We consider metal self-diffusion and choose vacancy concentration as our composition variable. Allowing the elastic moduli to vary with vacancy concentration via a simple rule-of-mixtures introduces a double-well shape to the Gibbs free energy density. A minimum at low vacancy concentration corresponds to the equilibrium value usually observed: about 7-8 orders of magnitude below the lattice site concentration at room temperature in aluminum. However, a second, deeper minimum also exists for vacancy concentration equalling the lattice site concentration. A large energy barrier exists between these states and, under most regimes of temperature, stress, electric field strength etc., the barrier is not surmounted. We have examined situations under which this barrier is reduced or made to disappear altogether. The implications for stability of the structure and for void formation are examined. The study is analytic and computational, and coupled boundary value problems have been solved in the regimes of interest.

[1] Lattice-based Micromechanical Continuum Formulation for Stress-driven Mass Transport in Poly- crystalline Solids, K. Garikipati, L. Bassman and M. Deal, J. Mech. Phy. Sol., 49, 1209-1237, 2001.
[2] Atomically-based Field Formulations for Coupled Problems of Composition and Mechanics, K. Garikipati and L. Bassman, in Materials Research Society Symposium, vol 653, pp Z9.6.1-Z9.6.6, 2001.