An Analytic Approach to the Radiation Fields of Tubular Halogen Lamp Arrangements in RTP Reactor Blocks
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INTRODUCTION The simulation of wafers heated by rapid thermal processing systems has been done by several authors. Kakoschke et al [1, 2] showed the appearence of spatially nonuniform temperature distributions on wafers during a rapid thermal process. Simulations based on ray-tracing and MonteCarlo methods by Tillmann et al. [3, 6] delivered homogemeity mappings which mostly imaged the practical results. Jensen et. al. [4] developed numeric models based on finite element methods and Kiether et.al. modelled a 3-zone RTP-system [5] with a thermal wafer model. Other publications deal also with chamber design [7] or with secondary effects like direct irradiation of the sample or sample irradiation after reflection from the mirror [8].
This publication describes more the basic physical properties of radiation sources, neglecting multireflections, convective influences and thermoconductivity effects. Starting with the principles of Beer's law a set of analytical lamp radiation formulas for pointlike and one-dimensional light source arrangements is derived. The results are applied to lamp configurations, which are used in rapid thermal processors. The following discussion compares crossed lamp arrangements with interlacing (non-crossed) lamp arrays.
MATHEMATICAL APPROACH Beers Law Assume a pointlike radiation source of power P, emitting its radiation equally in all directions. If a sphere with radius r around this source is put in a way, that the radiation source is located in its center, the intensity of the radiation on all points of the spherical surface is the same and the total
radiation power on the surface is P. Thus the radiation intensity I(r) can easily be expressed by: I(r) = P/(47r 2 )
327 Mat. Res. Soc. Symp. Proc. Vol. 389 ©1995 Materials Research Society
(1)
If the pointlike radiator is located in a distance h above the center of the xy-plane, the radiation intensity I(x,, y., z=0) on this plane can be derived. The situation is demonstated in figure 1: spot light (0-dimensional
(0, 0)
light source)
x,
(Xc, Yc)
y-Direct ion
Figure 1: Spotlight with distance h to the xy-plane
The intensity I(x,y) in W/cm 2 in a point (xc, yc) can be determined by use of the the geometrical theorem of Pythagoras:
sin(a) ~ I(xc, Yc) = 4r( P22• +x
(2)
4rt (h + x2 + y2)
I(xc, Yd)
P °h 4/:(2 + Xc2+ yc•2)3/2
(3)
Superposition of Spotlike Radiation Sources Arranged Equidistant in a Line Parallel to xy-Plane Let n be the number of pointlike lamps, P be the total power of all lamps and P/n be the power of one lamp. Furthermore let the lamp number k be located at (x V y , h) with: Xk= (2k-i). 1/ 2n and Yk = 0.1 shall be the length of the lamp cKain according to figure 2 below:
.•In
.l.-
\.1
Nd
Nd
I
h spot lamps
0
xl
x2
xc
Ixn
Figure2: A chain of spotlamps parallelto the x-axis 328
xAi x-Axis
The projected intensity distribution of this chain on the xy-plane can be determined from eq. 3: c
P- h kn~c = I 4En (h 2 + (Xc-C(2k-01)) 2 + Y2) 3/2
(4)
y~C
2n
t~h+(c~ k~l 7
The Radiation Characteristics
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