HOLE CONCENTRATION AND POINT CONTACTS 493 



The value of y can also be found for r = 0. For z = 0, r = 0, we have: 



v=^^"p-^' = ^G(Xp). (24A) 



The integral can be expressed in terms of integral sine and cosine func- 

 tions: 



G{k) = 2 f ^^^ ^ 2 T-cos^ fsi ^ - ") + sin k Ci k 



TT Jo X + X TT |_ \ 2/ 



(25A) 



If k is not too large, G{k) is nearly equal to F{k), so that y is approximately- 

 constant over the area of the disk. 



The total current flowing from the contact is found from integrating 

 kT^ip (dy/dz) over the disk: 



/^ = - kT,, yof f *"^°^"^ f " ' dl dr (26A) 



Jo Jo / -|- A 



JO / -f- A 



The integral can be evaluated with use of the general integral of Watson, 

 to give: 



I pa = -4pkTnpyo H(\p), (28A) 



where 



H{k) = f :^iM^^ =. -^ [cos k jm + sin k \\{k)l (29A) 

 Jo X -\- k I 



Using (23 A) for yo, vve have: 



I^a = -4pkTfjL,pAH(\p)/F(\p)]. (30A) 



Except for the factor H(\p)/F(\p), this expression for the current is 

 identical with (16A). This factor, which gives the effect of recombination 

 on the current, is plotted in Fig. 10. Recombination gives an increase in 

 current flow, but the effect is small for the normal rate of surface recom- 

 bination, which corresponds to ^ = Xp -^^ .035. 



APPENDIX B 



Calculation of Hole Flow for Arbitrary Hole Concentration 



In the text it was assumed that the concentration of holes was suffi- 

 ciently small so that the first term in the brackets of Eq. (26) could be 

 neglected in comparison with unity, yielding Eqs. (28) and (29). We give 



