Interferometric Measurements of Surface Properties during Laser Annealing
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ABSTRACT A simple interferometric method is presented which allows measurement of small vertical displacement of a surface heated by a laser beam. Calculations applied to a silicon crystal in the case of a c.w. laser show reasonable agreement with experiment. The method can be applied to assess surface temperature and thermal constants.
INTRODUCTION The problem of the determination of the temperature reached by a semiconduc tor surface irradiated with a laser is considered by using an interferometric method to measure the surface thermal dilatation. Application to the case of slow heating with a cw laser is here treated. The basic point is the fact that under a temperature increase the illuminated surface suffers a vertical displacement (here denoted as dilatation) which is the measured quantity. Unfortunately the relation between temperature increa se and dilatation is rather involved; however with some approximation a solution in closed form can be given which is rather good near equilibrium. The method can be used to derive also other interesting parameters of the material under study. THEORETICAL RELATION BETWEEN DILATATION,
TIME AND TEMPERATURE
The problem of heating and subsequent displacing of a thin semiconductor slab of thickness I illuminated on one side by a gaussian cw laser is treated. The laser intensity is written as 2 2 l(r) = Ilexp (- 2r /w* ) (1) where w. is the spot. A thermoelastic quasistatic formulation is for the displacement t is taken as V22 -> u--
used
(1).
The starting equation
-> 1 12v rotuu= a grad T
(2)
where cc =2a being
v
+ v 1 - 2v
the Poisson ratio, a
the thermal dilatation coefficient,
Mat. Res. Soc. Symp. Proc. Vol. 13 (1983) QElsevier Science Publishing Co.,
Inc.
and T the
218
temperature. Asolution is foundby using a method proposed by Goodier (1) which makes use of the thermoelastic potential 4)(xyzt) related to u by + all ay
D(D Dz
(3)
being u a particular integral of (2). Eq.(2) then writes V2ýD.'- 1+V 1-') ) T
(4)
being T the temperature distribution which inour case istaken as ist
2 Q.(l-R)W,2 ex-)(-at) exo(-r 2/(4X t+WO-)) f0 12c(4Xt + w2 -l.dz'exp(-(z-z') 2AXt) (5) 0 0
where Q,isthe energy incident per unit area at the center of the gaussian profile of radius w ,a = 2HX/lk, being H the heat transfer by free convection (which isa function7 of the geometry, the origntation and the temperature dif,.5.10-4 W CM-4 C (2)), R isthe reflectivity, ference; for surfaces inair H -X isthermal diffusivity, c isthermal capacity, and k isthermal conductivity. k14 1 1 If T-T. = AT
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