Modeling and simulation are a crucial aid for reducing development cycle time and costs in modern semiconductor technology. New modeling concepts increasingly require long-term research performed in an interdisciplinary manner, and new numerical methods and algorithms are needed to implement these concepts.
The manifold of physical phenomena inherent in manufacturing processes and the complex electromagnetics phenomena of interconnect structures require a large diversity of mathematical formulations. In this thesis I consider the numerical methods for solving the partial differential equations (PDEs) appearing in advanced diffusion and interconnect reliability models. Starting from the general finite element method I present discretization and linearization methods for non-linear PDE equation systems. Since the finite element method closely depends on the properties of the underlying mesh and used time step control strategy, their connections are investigated and solutions are proposed.
All discussed algorithms are eligible for the simulation in arbitrary three-dimensional domains. The capabilities of the presented numerical schemes are presented by simulation examples which at the same time motivate the three-dimensional approach. That is the case for the diffusion simulation in the presence of surface reactions and boundary conditions defined on explicit three-dimensional interfaces. The surface dynamics of the point defects has a strong impact on their overall behavior thus retarding or enhancing diffusion of dopants and determining the junction formation.
The material transport induced by electromigration and accompanying driving forces is, due to the structure and composition of modern interconnect layouts, a three-dimensional phenomenon. The barrier layers, applied in the contemporary interconnect copper based technology, determine the material flux and promote or prevent electromigration. Three-dimensional mechanical stress distribution, having its source in thermal mismatch between layout materials or local vacancy dynamics, plays a significant role too.
Apart from the simplistic and restrictive design rules, advanced, physically based simulation of electromigration phenomena, gives the opportunity to bring the ``driving forces'' to work, thus enhancing the reliability of the given interconnect structure.
A special focus of this thesis is the careful derivation and analysis of the numerical schemes for the PDEs describing relevant physical phenomena and determining appropriate mesh and time step controlling rules.
In this work complex diffusion models are handled by using advanced, finite element based, numerical methods. Algorithms for solving of these diffusion models are constructed and implemented in a software tool FEDOS (Finite Element Diffusion and Oxidation Simulator). The applied discretization and linearizations schemes are well mathematically founded. Interface reactions are modeled with the contemporary models which are integrated in the simulator in a rigorous way. Three-dimensional simulations have been performed for several diffusion process steps relevant for the processing of state-of-the-art semiconductor devices. The simulation results demonstrate the physical plausibility of the applied models and numerical methods as well as the necessity of three-dimensional simulations.
In this thesis I discuss different models and approaches dealing with electromigration reliability problems in modern interconnect layouts. Generally, a two stage approach for the electromigration simulation is adopted. The physics of the time period until the void nucleation is modeled as dynamic bulk vacancy phenomena and subsequent void evolution as void/metal interface material transport phenomenon. The numerical schemes to handle of the governing equations are presented together with appropriate mesh adaptation schemes. The predictive capability of the considered models is demonstrated by simulations which are carried out on different realistic interconnect structures.