The MacWafer™ code computes gravitational and thermal stresses in silicon wafers and uses these results to determine the maximum allowable temperature variation across a wafer, maximum processing temperatures, and maximum allowable heating and cooling rates. This information is of particular interest in the case of processing 300-mm wafers coupled with fast ramp technologies. The program runs interactively on Apple Macintosh, IBM PC, and PC clones, and workstation computers as well. Execution time is typically about 20 seconds on a Motorola 68040 processor operating at 33 MHz.

Gravitational stresses and displacements are first calculated based on the user's input of a wafer support geometry. This may consist of one or more continuous rings or up to 20 arbitrarily placed point supports. The maximum allowable processing temperature is then computed by comparing calculated gravitational stresses with the temperature-dependent strength of the wafer. One of the three published models of the strength may be selected. The difference between wafer strength and the gravitational stress is used to determine the allowable thermal stress and allowable temperature variation across the wafer. Finally, a model of radial heat transfer in a batch furnace is used to compute the maximum heating or cooling rate as a function of the allowable temperature difference and the user's inputs of wafer spacing and maximum available heater power.

Outputs to the screen include all stress components and vertical wafer displacements, as well as tables of maximum stresses and maximum heating and cooling rates as a function of temperature. All inputs and outputs may be directed to user-specified files for further processing or for display via user-supplied graphics software.

Gravitational stress and wafer displacements are computed by solving the fourth-order elliptic partial differential equation governing elastic deformation of thin plates. The particular solution for a specified support geometry is constructed by superposition of general Fourier-series solutions applicable to ring and point supports. A nonlinear equation solver is used to determine the fraction of the wafer weight carried on each support element, including cases in which the wafer may not contact all supports. Thermal stresses are also determined analytically by evaluating known integral solutions for the case of an axisymmetric temperature field that varies quadratically with distance from the wafer's center. For the special case of a batch furnace, the radial temperature variation across the wafer during heating and cooling is related to the ramp rate based on an analytical solution to the problem of radial heat transfer. This solution accounts for both conduction through the silicon and radiation between opposing wafer faces, as well as for direct radiant exchange between wafer surfaces and the furnace walls. The total stress is obtained by tensor addition of local components of the thermal and gravitational stresses.

MacWafer™ is the only existing computer code that is specifically written to analyze stresses in silicon wafers. Both the user interface and solution procedures are highly specialized and optimized well beyond those available in general-purpose finite-element and finite-difference solid-mechanics codes. As a result, execution times for all MacWafer routines are short enough to permit enjoyable interactive use on personal computer platforms. In addition, the interactive interface is sufficiently self-explanatory that there is no need for a user's manual. All required code inputs are displayed on the screen, including initial default values that may be interactively altered by the user. Help, notice, or warning messages are displayed whenever input problems arise. The code is also unique in automatically performing many of the iterative trial-and-error searches required to yield the results most needed by equipment designers and process engineers. For example, the code directly computes maximum allowable temperatures and maximum heating and cooling rates, rather than simply indicating whether or not plastic deformation occurs for a given set of process conditions.

This software was developed by Bob Nilson and Stewart Griffith of Sandia National Laboratories, Livermore, CA, under DOE contract no. DE-AC04-94AL8500. Information regarding licensing of the software can be obtained from Sandia's web page:, or by contacting C.V. Subramanian, Manager of Licensing, at (510) 294-2311.

Photonics Tech Briefs Magazine

This article first appeared in the December, 1998 issue of Photonics Tech Briefs Magazine.

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