Thermal Analysis of CW Laser Annealing Beyond the Melt Temperature
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THERMAL ANALYSIS OF CW LASER ANNEALING BEYOND THE MELT TEMPERATURE
S. A. KOKOROWSKI, G. L. OLSON AND L. D. HESS Hughes Research Laboratories, Malibu, CA 90265
ABSTRACT We present a thermal analysis which treats the problem of melting as it occurs during cw laser heating. Analytical expressions for sample temperature distributions are derived, and calculated results are compared to experimental measurements.
The general application of cw laser processing techniques to semiconductor materials involves rapid localized heating to high surface temperatures and, in some cases, controlled melting of thin surface layers [1,2,3]. Although the thermal analysis of laser heating has been treated previously [4,5,6], these treatments have been limited to the solid state regime, due to the difficulties involved with solving the nonlinear thermal diffusion equation which results when melting effects are included. In this article we present a theoretical model of cw laser heating which accounts for the abrupt changes in optical and thermal properties of semiconductor materials that occur upon melting. We derive analytic solutions to both the one dimensional and three dimensional time independent thermal diffusion equations. Following derivation of the analytical expressions, sample calculations using the three dimensional solutions are presented and compared with experimental measurements. The theoretical model is based on the physical concept that laser induced melting begins by the formation of a very thin liquid film at the center of the laser spot. As the depth of this liquid increases due to increased laser power, thin film optical interference effects cause the local reflectivity of the sample to increase continuously from the lower solid-state value to the higher liquid-state value. This increase can be easily calculated using classical plane wave optical techniques [7], and is shown graphically in fig. 1, for the 0.8
0.7 Fig.
1.
Reflectivity of a liquid Si film on a solid Si substrate
0.6
(•=5000A)
(A
=5000)
LUINCIDENT aL
_
0.5
I-
LIGHT
I REFLECTED oLIGHT LIGH
0.4
AIR
Si (LIQUID)
Si (SOLID)
0.3 0
200
400
600
LIQUID FILM THICKNESS, A
800
140 case of an argon laser beam (X=.5 1Am)normally incident on a sample of crystalline Si having a uniform thin liquid surface film. Since a Gaussian shaped laser beam will not produce a liquid film of uniform depth the optical reflectivity will vary spatially across the sample surface with a profile similar to the liquid depth profile. The local reflectivity will have a maximum value at the beam center where the depth is greatest and will decrease towards the solid state value at increasing radial distance, where the melt depth decreases. This situation can be described mathematically by first assuming that the reflectivity variation across the laser spot will always have the functional form, 2 R = RO + R -exp () where R. is the solid state value and R, is the increase from this value at the beam center. The variable r is the radial distance from beam center
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