CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 589 



The close connection between M defined by (10.26) and h defined by 

 (7.17) is apparent. Thus in (7.17) when r = L, exp[-47rrW/3] is e~\ 

 and for larger values of r this exponential quickly forces the convergence 

 of the integral. Therefore the values of h and M will be almost equal. 

 This is not surprising since they are meant to be the same thing, i.e., 

 the average concentration, c(oo), of donors at infinite distance in the 

 equilibrium atmosphere of an acceptor. Both quantities are computed 

 so as to conserve charge in this atmosphere. 



At large values of N, M proves to be much smaller than A^ so that 

 (10.25) reduces to (10.14), validating the crude treatment, for r in (10.24) 

 is obviously the relaxation time. This is easily seen by writing 



-^ = -4,rrV*(r) = ^^« .""^ (10.27) 



at kkT 



from which one derives by integration 



n = M + (AT - M)e~"'' (10.28) 



According to (10.28) at ^ = 0, n = A", the correct initial density for 

 unpaired ions. At ^ = 00,72 = M, also the correct density, i.e., the 

 density at large values of ?-, when equilibrium is achieved. Obviously r 

 plays the role of the relaxation time, since by differentiation of (10.28) 



din - M) _ {n- M) ^^^^9) 



dt T 



which is to be compared with (10.2) and (10.3). 



Values oiM can be computed using formulas (9.10), (9.11), and (9.12) 

 and Figs. 16 and 17 since the integral in (10.26) is one of the i integrals 

 I'lg. 18 shows some values of M, computed in this way for the tempera- 

 tures 206°, 225°, 250°, and 300°K, for a semiconductor where the value 

 of a = 2.5 X 10~^ cm, k = 16, and q = 4.77 X 10"^" statcoulombs. The 

 plots are of M versus N. Note that the values of M are generally much 

 less than A", the disparity increasing with lower temperatures and larger 

 A. 



It is also possible to calculate t for the above system in its dependence 

 upon A" and T. To do this the value of Z)o must be known as a function 

 of temperature. Fuller and Severiens have measured the diffusivities of 

 lithium in germanium and silicon down to about 500°K. These data plot 

 logarithmically against \/T as excellent straight lines. In Fig. 19, we 

 show an extrapolation of the line for lithium in germanium down to the 

 neighborhood of 200°K. From this figure it is possible to read values of 



