Modeling and Simulation
This chapter is devoted to recent developments in mathematical modeling and computer simulation of copper electrodeposition. We focus our attention on continuum models and kinetic Monte Carlo simulations for shape evolution and the effects of additives on
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Modeling and Simulation Yutaka Kaneko
This chapter is devoted to recent developments in mathematical modeling and computer simulation of copper electrodeposition. We focus our attention on continuum models and kinetic Monte Carlo simulations for shape evolution and the effects of additives on copper deposition, especially the filling of small features in microelectronics. The modeling, mathematical treatments, and simulation results are reviewed with brief summaries of efficient numerical algorithms. Fast computing and prospects of simulation research are also discussed.
4.1 Introduction Copper electrodeposition has attracted a great deal of attention since IBM announced the replacement of conventional vapor deposition of aluminum with copper electrodeposition for the production of LSI interconnects [1]. The dual damascene process is now a central technique for the fabrication of threedimensional (3D) LSI circuits. An important requirement for the success of this process is the capacity to fill submicron features such as via holes and trenches completely without voids or seams. This process is called ‘‘superfilling’’ or ‘‘superconformal filling.’’ It has been found experimentally that superfilling is achieved by the synergistic effects of different kinds of additives, and the optimal deposition conditions have been explored. Mathematical modeling and numerical simulations are indispensable means of research to understand the underlying chemistry from a microscopic point of view and to determine the optimal conditions for practical applications. There are two types of modeling for copper electrodeposition. The first type is based on
Y. Kaneko (&) Department of Applied Analysis and Complex Dynamical Systems Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan e-mail: [email protected]
K. Kondo et al. (eds.), Copper Electrodeposition for Nanofabrication of Electronics Devices, Nanostructure Science and Technology 171, DOI: 10.1007/978-1-4614-9176-7_4, Springer Science+Business Media New York 2014
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‘‘continuum models’’ in which the physical quantities are expressed in terms of continuous variables. The basic equations are partial differential equations. The second type is ‘‘molecular simulation’’ such as Monte Carlo (MC) and molecular dynamics (MD) computations in which ions and molecules are treated as particles. In this chapter, we overview the recent developments of these two types of simulation for copper electrodeposition to understand the present status of sophistication. This chapter is organized as follows. The next section is the brief review of the numerical simulations based on continuum models. Section 4.2.1 is devoted to the mathematical models for superfilling. The basic ideas and mathematical formulas of the diffusion-consumption theory [1–5], the recent theory of curvature enhanced accelerator coverage [6–14], and the theory based on time-dependent transport [15, 16] are described. We then review the nucleation theory in t
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