626 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



yields F = 0. In the other extreme No = 0.9 A'' a so that we get 



l-T 

 7 = ^ ^n 10 (D23) 



Q 



This therefore is the largest value which Vo may assume in our case. 

 Inserting the expression in D21 in place of Vo we end with the relation 



10 < ^ (D24) 



Thus provided that in the distribution being considered 



Xmin > 3.5^ (D25) 



there will be no space charge anywhere. 



At high temperatures 0.1 Na may be less than rii . Under these condi- 

 tions (D24) should be replaced by 



12a. ^ Amrn ^j^26) 



rii P 



and in the limit that rii becomes very large it is obvious that (D26) will 

 always be satisfied. The rule to be enunciated for the cases we shall be 

 interested in is the one given in section XI, i.e. that no space charge will 

 exist provided that X min is no less than order, /. 



Appendix E 

 calculation of diffusivities from conductances of diffusion 



LAYERS 



In this appendix equation (11.12) will be derived. In the first place 

 we note that the dependence of Nd on position x, and time t, will be of 

 the form Nc{x/\/t) at any stage of the diffusion process. This results 

 from a theorem due to Boltzmann^^ that when the dependence of D upon 



X and t is of the form D(Nd), i.e., the dependence is through Nd , and a 

 semi-infinite region extending from x = to a: = oo is being considered, 

 then, in the case of plane parallel diffusion, the only variable in the prob- 

 lem will be x/\/}. 



Although the wafers considered in Section XI are of finite thickness d, 

 the stages of diffusion investigated are such that the two regions of loss 

 near the surfaces have not contacted each other. As a result the system 

 behaves like two semi-infinite regions backed against one another, and 

 the preceding arguments hold. The conductance 2, defined in section 



XI will be proportional to the integral of the product of the local carrier 



1 



