CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 627 



density by the local mobility. Thus 



S = CO / ix(x, t)[NA - ND{x,t)]dx 



(El) 



where co is a proportionality constant and n(x, t) is the local mobility. 

 An upper limit of d/2 rather than d is used because of symmetry. The 

 local mobility will vary because No , and therefore the local density of 

 charged impurity scatterers, varies. Let No be the initial uniform den- 

 sity (before any diffusion out) of donors, and write (El) as 



pan 



S = CO n(x, t)[NA - No + No° - Nn(x, t)] dx 



''0 

 = CO / IX{X, t)[NA. - Nd] dx + CO / li{x, t)[ND 



Jo Jo 



(E2) 



- NdCxjOj^x 



The second integral on the right of (E2) is given the upper limit co , 

 because in the experiments we wish to perform No — No becomes zero 

 long before x reaches d/2. 



Now in the first integral on the right of (E2) we may set fjL(x, t) equal 

 to the constant value no , which it assumes in the bulk of the wafer, be- 

 cause the breadth of the depletion layer near the surface (in which 

 (i(x, t) departs from juo) is small compared to d/2. The same thing can- 

 not be done in the second integral since the integrand vanishes beyond 

 the depletion layer and the total contribution comes from that layer. 

 We thus obtain 



2 = com''(N^ - Nz,°) d/2 



+ C0 



X 



"VV?, 



.^"° - ^° ivi)] "" 



(E3) 



In the integral in (E3) both /x and No are represented as functions of 

 x/-\/i, the latter because of what has been said above, and the former, 

 because it is a function of the latter. Defining 



V = x/2\/Dt (E4) 



in which D is constant, and substituting in (E3) gives finally 



2 = co/Xo(N^ - N/)f//2 + 2o^\/Di f m('')[Nz,° - Nx,(^)]rf^ (E5) 



Jo 



