Mathematical modeling of vapor-plume focusing in electron-beam evaporation
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I. INTRODUCTION
THIN films made by electron-beam physical vapor deposition (EBPVD), shown in Figure 1, play an increasingly important role in a wide variety of products and fields, from electronics to thermal-barrier coatings for turbine blades to metal-matrix composites. However, the mechanics of vapor transport are poorly understood, in part because rarefied gas behavior is very different from that of more familiar fluids. In particular, the flux of atoms evaporating from a surface follows the well-known cosine distribution, but in highflux processes such as electron-beam evaporation, collisions between evaporant atoms actually lead to a focusing of the vapor plume toward the surface normal, with the resulting flux distributed as cos2 or even cos3 .[1–6] This result is completely different from the intuitive expectation that more collisions will lead to dispersing of the plume, and so it is conceptually difficult to identify collisions as the cause of this focusing. Understanding the dynamics of the plume is important to the prediction of the distribution of gas species in the chamber and, in turn, the prediction and control of the deposition rate and thickness uniformity across substrates from design parameters. In the past, some have chosen to model the focusing using a cosine distribution at a “virtual source” located directly above the actual source (Reference 7, p. 80). While this is somewhat useful for a single flat substrate parallel to the source, it is not helpful for any other geometry and gives a less accurate fit to the profile than a cosn distribution. The extent to which a uniform gas behaves like a continuous fluid is described by the Knudsen number, which is the ratio of the mean free path to the chamber size (/L), with the mean free path for a uniform gas given by
A. POWELL, Assistant Professor, is with the Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139. P. MINSON, Production Engineer, is with Intel Corporation, Rio Rancho, NM 87124. G. TRAPAGA, Professor, is with the Laboratorio de Investigacio´n en Materiales del CINVESTAV-IPN, Unidad Queretaro, Fracc. Real de Juriquilla, C.P. 76230, Queretaro, Qro., Mexico. U. PAL, Professor, is with the Department of Manufacturing Engineering, Boston University, Boston, MA 02446. Manuscript submitted July 24, 2000. METALLURGICAL AND MATERIALS TRANSACTIONS A
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1 2 n 冪2
[1]
where is the collision diameter and n is the number of molecules per unit volume. When the Knudsen number is very large (i.e., the density is very low), molecules travel in straight lines across the chamber, with a low probability of a collision interrupting that trajectory. At the other end, when the Knudsen number is very small (i.e., a high density), collisions are so frequent that one may treat the vapor phase as a continuum fluid. But between these two extremes, there is a wide range of Knudsen-number values for which neither model gives an accurate representation. Figure 2 shows the Knudsen numbers given by ty
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