RESEARCH PAPERS: Heat Transfer in Manufacturing

High Knudsen Number Physical Vapor Deposition: Predicting Deposition Rates and Uniformity

[+] Author and Article Information
Chetan P. Malhotra1

 Tata Research Development and Design Centre, India, 54/B, Hadapsar Industrial Estate, Pune, 411013, Indiachetan.malhotra@tcs.com

Roop L. Mahajan2

Department of Mechanical Engineering, University of Colorado at Boulder, Campus Box # 427, Boulder, CO 80309-0427mahajan@spot.colorado.edu

W. S. Sampath

B120 Engineering Research Center, Colorado State University, Fort Collins, CO 80521sampath@engr.colostate.edu


Corresponding author. Also International Centre for Science and High Technology, Trieste, Italy; University of Colorado at Boulder.


Current address: Chair Professor and Director, ICTAS, Virginia Tech, Blacksburg, VA 24061.

J. Heat Transfer 129(11), 1546-1553 (Jul 21, 2006) (8 pages) doi:10.1115/1.2712855 History: Received April 11, 2006; Revised July 21, 2006

The problem of predicting deposition rates and film thickness variation is relevant to many high-vacuum physical vapor deposition (PVD) processes. Analytical methods for modeling the molecular flow fail when the geometry is more complicated than simple tubular or planar sources. Monte Carlo methods, which have traditionally been used for modeling PVD processes in more complicated geometries, being probabilistic in nature, entail long computation times, and thus render geometry optimization for deposition uniformity a difficult task. Free molecular flow is governed by the same line-of-sight considerations as thermal radiation. Though the existence of an analogy between the two was recognized by Knudsen (1909, Ann. Phys., 4(28), pp. 75–130) during his early experiments, it has not been exploited toward mainstream analysis of deposition processes. With the availability of commercial finite element software having advanced geometry modelers and built-in cavity radiation solvers, the analysis of diffuse thermal radiation problems has become considerably simplified. Hence, it is proposed to use the geometry modeling and radiation analysis capabilities of commercial finite element software toward analyzing and optimizing high-vacuum deposition processes by applying the radiation-molecular flow analogy. In this paper, we lay down this analogy and use the commercial finite element software ABAQUS for predicting radiation flux profiles from planar as well as tube sources. These profiles are compared to corresponding deposition profiles presented in thin-film literature. In order to test the ability of the analogy in predicting absolute values of molecular flow rates, ABAQUS was also employed for calculating the radiative flux through a long tube. The predictions are compared to Knudsen’s analytical formula for free molecular flow through long tubes. Finally, in order to see the efficacy of using the analogy in modeling the film thickness variation in a complex source-substrate configuration, an experiment was conducted where chromium films were deposited on an asymmetric arrangement of glass slides in a high-vacuum PVD chamber. The thickness of the deposited films was measured and the source-substrate configuration was simulated in ABAQUS . The variation of radiation fluxes from the simulation was compared to variation of the measured film thicknesses across the slides. The close agreement between the predictions and experimental data establishes the feasibility of using commercial finite element software for analyzing high vacuum deposition processes.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

Geometry for verifying Knudsen’s formula for free molecular flow through a long tube

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Figure 2

(a) Schematic of configuration for deposition from two planar sources and (b) Radiation flux distribution for D∕H=0.75

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Figure 3

Flux distribution from two coplanar sources. Continuous lines represent expected normalized film thickness variation along centerline of sources given in (30) while open markers represent results of radiation simulations in ABAQUS .

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Figure 4

Geometry for simulation of angular distribution of radiation flux from short tubes

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Figure 5

Comparison of normalized angular distributions of radiation flux from short tubes with aspect ratios 1.15, 3.03, and 5.63 from ABAQUS (continuous lines) with experimental measurements of Stickney (28) (solid dots). The open dots represent the cosine law which correspond to L=0.

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Figure 6

Schematic of experimental setup

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Figure 7

Contours of normalized radiation fluxes corresponding to the experimental setup of Fig. 6 simulated using ABAQUS

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Figure 8

Comparison of measured normalized film thickness at the center of each slide with corresponding normalized radiation fluxes from ABAQUS




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