A fitted finite element method for the numerical approximation of void electro-stress migration
Introduction
Microelectronic circuits contain thin lines of aluminium alloy, that make electric contact between neighbouring devices possible. These lines are passivated with a layer of oxide at large temperatures, and during the cooling process large stresses are induced. As the dimensions of microchips are reduced further and further, and since interconnects always contain small voids or cracks, it is of great interest to investigate the physical mechanisms that impede such a reduction, due to mechanical failures in the lines induced by the motion of the cracks. The problem analysed in this paper involves the evolution over time of voids in a conducting metal line where three different contributions to the drift of the voids are present: the surface tension, the electric field and the elastic energy. This phenomenon is known as electro-stress migration; for further details see, e.g., [36], [14], [3], and the references therein.
As the height of interconnect lines is much smaller than the dimensions of the base, voids generally fully penetrate the conducting material. Hence it is common to consider a two dimensional model for void electro-stress migration, and this is the approach that we are going to pursue in this paper. In addition, for ease of exposition, we assume that the interconnect line is given by a rectangular solid. The electric field is induced in the line by prescribing the voltage on its vertical boundaries, while the displacement field is induced by prescribing the stresses on its four boundaries.
In this paper, based on our previous work in [31], we introduce a novel front-tracking, fitted finite element method for the approximation of void electro-stress migration. The main difference to the approximation presented in [31] is that here we consider the fitted approach, which means that the interface mesh is always part of the boundary of the bulk grid. Moreover, we also include the effect of stress-migration into the model. As an aside we note that our method inherits the good interface mesh properties from the approximation in [31]. In particular, the vertices on the discrete interface equidistribute asymptotically so that no reparameterisation of the discrete interface is necessary in practice.
The paper is organised as follows. In Section 2 we give a mathematical description of the problem of void electro-stress migration that we are interested in. We also give a brief overview of the different numerical methods applicable to this problem. In addition, we highlight the differences between the fitted approach presented in this paper and the unfitted approach previously introduced by the authors in [31]. Section 3 contains a detailed description of our proposed finite element approximation. In Section 4 we discuss possible solution methods of the algebraic system of equations arising at each time level. In addition, we present details on the bulk mesh smoothing strategy. Finally, in Section 5 we perform a convergence experiment for a test case in which the exact solution is known, and we present various other examples of the application of our numerical method.
Section snippets
Problem formulation
For the formulation of the governing equations we closely follow the presentation in [31], see also [3]. Let , where , be the domain that contains the conductor. We denote the boundary of Ω with ∂Ω. At any time , let be the boundary of the void inside the conductor Ω. Then and denotes the conducting region (see Fig. 1). Now the evolution of the interface , which represents the void boundary, is given by
Finite element approximation
We begin with the finite element approximation for quantities defined over the bulk mesh. Let be a partitioning of into possibly variable time steps , . We set . Let be a partitioning of , a polygonal approximation of , into open disjoint triangles. Let be the inner boundary of , so that . We can now define the standard finite element space of piecewise linear functions:
Solution method
Due to the special structure of the system (17a), (17b), (17c), (17d), the equations for , and decouple. In practice, we can find the unique solution to (17a), (17b), (17c), (17d) as follows. First we find such that where is the standard stiffness matrix for the Laplacian on , i.e. where are the basis functions of the unconstrained finite element space . In the above we have ignored the effect of the
Numerical experiments
We implement our finite element approximation within the framework of the C++-based software DUNE, see [11], [10], and we employ the Alberta grid manager, see [34]. Unless stated otherwise, we use uniform time steps , , for all the numerical experiments in this section. As we will compare our numerical results to the phase field computations in [9], [3], we also fix . When stress-migration is considered, we assume that (6) holds and put throughout this section. On
Conclusion
We have presented a fitted front-tracking method for the approximation of electro-stress migration using parametric finite elements. The main properties of our method are that we can prove an equidistribution property for the vertices on the discrete interface, and that in the absence of external forces the scheme can be shown to be unconditionally stable. We note that in practice, in all our computations, the method is also stable in the presence of forces due to the applied electric field and
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