Theory and Computer Simulation of Grain-Boundary and Void Dynamics in Polycrystalline Conductors
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Mat. Res. Soc. Symp. Proc. Vol. 391 01995 Materials Research Society
(void) surfaces. The general framework for progress in this direction is available [28,29]. Computationally, a robust tool is provided by the finite-element method [30], while the systematic search of the parameter space for engineering applications can be accomplished by nonlinear continuation techniques [31]. In this paper, we present the basic elements for self-consistent modeling of diffusional mass transport in heterogeneous solids and address some critical problems regarding grain-boundary and void dynamics in interconnect lines. Stress-induced formation and diffusional growth of intergranular voids in passivated bamboo-structure conductor lines and the morphological evolution of transgranular voids under electromigration conditions are analyzed. In all cases, the theoretical results are discussed in the context of recent experimental data in aluminum conductor lines. MASS TRANSPORT MODELING IN HETEROGENEOUS SOLIDS Diffusional mass transport is among the dominant mechanisms of microstructure evolution in heterogeneous solids under the action of external fields. The heterogeneous solid of interest in interconnect problems is a thin-film polycrystalline conductor, where microstructural complexity arises from the presence of grain boundaries, internal free surfaces (voids), and heterophase interfaces. We outline below the general formulation of the problem and its computational implementation for a monatomic conductor. Diffusional fluxes are determined by the chemical potential A(s) of the atoms along an interface, either a free surface or a grain boundary, expressed by [32] IL, = po - Q q:fifi +i r(s) Q+
:
2,
(1)
where p0 is the chemical potential for a flat stress-free interface, 0 is the atomic volume. ii is the unit vector normal to the interface, a and c are the stress and strain tensors, respectively, y is the interface free energy, and K(s) is the local curvature. The elastic strain energy term in equation (1) is of second order in strain and can be neglected in most cases. The interfacial flux J8 is proportional to the gradient of /(s)
=
(2)
k
where D, is the interfacial diffusion coefficient, 6,/0 is the number of atoms per interfacial unit area, kB is Boltzmann's constant, and T is the absolute temperature. The rate of interface propagation v,, is given by the continuity equation [33] as
19t vn = Oun
Q VS.*J8)
(3)
where un is the local diffusional displacement normal to the interface. In the grains of f cc metals, such as Al, diffusion is governed by the vacancy mechanism. The evolution of the vacancy field is given by the continuity equation as 09cv Jv = -DvVcv, (4) 09 - -vV Jv; at
where cv is the bulk vacancy concentration, Dv is the vacancy diffusivity in the grain, and Jv is the corresponding vacancy flux. At the interfacial region, such diffusional fluxes induce inelastic displacements uv normal to the interface given by duv dv - Q[Jv. fi]. (5) dt 152
The boundary conditions for the vacancy diffusion problem in t
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