HOLE CONCENTRATION AND POINT CONTACTS 481 



in which Ip is the total hole current. The boundary condition (30) gives 

 the relation between Ip and Pb'. 



pb = pi-{- {Ip/lirkTiXpn). (33) 



Since the equations are linear, an equation of the form {i3>) applies to 

 the hole current due to the added holes as well as to the entire hole 

 current. For the former we have: 



Pba = Pa+ iIpa/2TrkTHj,n), (34) 



which is equivalent to Eq. (4). 



In the derivation of Eq. (34) we have neglected recombination at the 

 surface as well as in the interior. In the Appendix we give a solution for a 

 contact in the form of a circular disk and assume that recombination 

 takes place at the surface. The hole concentration then satisfies Laplace's 

 equation subject to more complicated boundary conditions at the surface. 

 The results are not significantly different from those of the simplified 

 model.' 



Equation (34), or rather its equivalent, Eq. (4), was used in the deriva- 

 tion of Eq. (12) for the floating potential, Vcf . If this value for Vc/ 

 is inserted into Eq. (15), an equation relating the conductance directly 

 with the added hole concentration is obtained: 



G = Go+ {ae'aVaA^Pa/^kT). (35) 



This expression may be simplified by substituting for a from Eq. (6): 



G = Go + a^HpeApa/n. (36) 



By using the expression for the normal conductivity: 



Co = bnpeno, (37) 



the conductance can be given in the form: 



G = Go + {a(3aoA/bn)(pa/m). (38) 



If (To is in practical units (mhos/cm), G is in mhos. 



We shall compare (38), which gives a linear variation between G and Pa, 

 with experimental data of Pearson^^ and of Suhl. The arrangement used 



' In the applications, these equations are applied to situations in which the contact is 

 on a germanium filament and there is a flow of current along the length of the filament in 

 addition to the flow to the contact. A question may arise as to whether it is justified to 

 neglect the filament current when discussing flow to the contact. There is no difficulty 

 as long as pa/m is small compared with unity because the equations are then linear and 

 the solution giving the flow to the contact can be superimposed on the solution giving the 

 flow along the length of the filament. The neglect of the filament current cannot be rigor- 

 ously justified in case pa/m is large, as is assumed in the calculations of Appendix B. It is 

 not believed, however, that the exact treatment would yield results which are significantly 

 different. 



'" See reference 3, p. 356 and Fig. 6. 



