6.152J3.155J Dopant or impurity concentration VS. Position assumptions: no solid state diffusion, perfect liquid mixing 6.12J/3.155J Microelectronic pror CZ growth Float zone th cts L S LL S [CL()-Cs(x)]dr=dC,(x[L-r ICo-kC,()dx=dC,(r)l, C(xl-k]dr=dC(x[L-x dr. cc (r)dCL(r) Co-kC(r C2(x) C Cs(x)=kCo Cs(x)=Col1-(1-k) Dopant/inpurity concentration VS. Position 6.12J/3.155J Microelectronic processing C(x)=kC「L L-x」 c(x)=C[1-(1-k)xp(-kx/6 CZ g Float zone growth k=0.l,L=1 k=0.l.l=0.L CsCo6.152J/3.155J 14 27 Nov.26 , 2003 6.12J / 3.155J Microelectronic processing Dopant or impurity concentration vs. Position assumptions: no solid state diffusion, perfect liquid mixing CZ growth C x=0 x=L S L Cliq = C0 CL (x) - CS [ ] (x) dx = dCL (x)[ ] L - x CL (x) 1[ ] - k dx = dCL (x)[ ] L - x [ ] 1- k dx L - x 0 x Ú = dCL (x) CL (x) C0 CL (x ) Ú CL (x) = C0 L L - x È Î ˘ ˚ 1-k CS (x) = kC0 L L - x È Î ˘ ˚ 1-k C0 - kCL [ ] (x) dx = dCL (x)l0 dx l 0 0 x Ú = dCL (x) C0 - kCL (x) C 0 CL (x ) Ú x l0 = -1 k C0 - kCL [ ] (x) C0 CL (x ) - kx l 0 = ln C0 - kCL (x) C0 (1- k) È Î Í ˘ ˚ ˙ C0 CL (x ) CS (x) = C0 1-(1- k)exp(-kx /l0 [ ]) Float zone growth C x = 0 lo x=L C0 S L S C0 28 Nov.26 , 2003 6.12J / 3.155J Microelectronic processing CZ growth CS (x) = kC0 L L - x È Î ˘ ˚ 1-k x=0 x=L 10 1.0 0.1 k = 0.1, L = 1 Float zone growth CS (x) = C0 1-(1- k)exp(-kx /l 0 [ ]) k = 0.1, l = 0.1L Cs/C0 Dopant/inpurity concentration vs. Position