FLOW OF HOLES AND ELECTRONS IN SEMICONDUCTORS 1209 



prescribed Ip and (?«, , or alternatively as limiting the possible choices of 

 Ip and (Poo for prescribed inner electrode radius. Had we chosen the con- 

 stants Q, R and 5 so as to obtain prescribed values of (P and D at a pre- 

 selected electrode radius tq , restrictions analogous to (172) on the maximum 

 radius would appear. 



For the case ^4 = the restriction analogous to (172) is 



Since the bracketed factor is positive, (173) provides no restriction for 

 Ip > 0, but for Ip < establishes a minimum radius of the kind just dis- 

 cussed. 



For L,A^O, the analog of (172) and (173) is 



Z/1 



-1 (^I^\ _ .-1 ( kT\N\ \ (174) 



\NeA) \ NeA ) 



r > 



where A and L are given by (153) and (154) and K~^ denotes the inverse 

 function of A — i.e., 



A-i[A(x)] ^ X 

 or 



A-HA) = A + In I A - 1 I . 



Equation (174) is a minimum radius restriction of the same kind as those 

 obtaining for yl = 0, and Z = 0, but the relationship between the minimum 

 radius tq and (?«, , Ip and In is considerably more complicated than in the 

 more degenerate cases. 



It will be noted that the relation 



eN A \ A ) 



(with I, B, M given in terms of (?« , /p , h by (152), (153), and (154)) deter- 

 mines which function (Ai , A2 , or A3) is to be used for A in any given case, 

 because any assigned value (p^ 0) is taken on by one and only one of 

 (Ai , A2 , A3). 



If surface recombination is negligible as well as interior recombination, 

 this spherically symmetric solution is of use in the study of "point" con- 

 tacts on a plane surface of a semiconductor. [Fig. 3 and Ref. 2.] 



The results of this section can easily be duplicated for any other choice 

 of the harmonic function 3C to obtain a great variety of specimen solutions. 



