472 BELL SYSTEM TECHNICAL JOURNAL 



assumed to be negligible so that the lines of current flow are radial. The 

 spherical symmetry of the resulting problem simplifies the mathematics. 

 A calculation is given in an Appendix for a model in which the contact is 

 a circular disk and recombination takes place at the surface. The latter 

 does not give results which are significantly different from the simplified 

 model. 



Figure 1(a) applies to the case in which the hole concentration deep in 

 the interior has its normal or thermal equilibrium value, p^. The sub- 

 script zero is used to denote values which pertain to this situation. Of a 

 voltage Vp applied to the contact, a part Vc occurs across the space- 

 charge barrier layer of the contact and a part Vi occurs in the body of 

 the semiconductor. Thus V p represents the voltage of the contact and V i 

 the voltage in the semiconductor just outside the barrier layer, both 

 measured relative to a point deep in the interior. It should be noted that 

 Vp does nol include the normal potential drop which occurs across the 

 barrier layer under equilibrium conditions with no voltage applied. In 

 the examples with which we shall deal in the present memorandum, the 

 spreading resistance is small compared with the contact resistance, so 

 that Vi is small compared with V p. Obviously, 



Fp = F, + Vi. (1) 



When a current is flowing to the contact the hole concentration, pbo, 

 measured just outside of the barrier layer, differs from the concentration 

 deep in the interior, pa. It is the concentration gradient resulting from 

 the difference between pm and />o which produces a flow of holes from the 

 interior to the contact. In the forward direction, />bo is larger than /»o; 

 in the reverse direction, pm is less than p^. 



The total current, /o( Fc), flowing across the contact includes both elec- 

 tron and hole currents. It will not be necessary to distinguish between 

 these two contributions to the normal current flow across the barrier 

 layer in the subsequent analysis. 



Figure 1(b) applies to the case in which the hole concentration deep in 

 the interior has been increased to />o + pa by adding a concentration Pa 

 to the normal concentration, />o. The concentration just outside the barrier 

 layer is increased to />m + pba- In addition to the normal current, h{V^, 

 flowing across the contact, there is an additional current of holes resulting 

 from the added hole concentration, pba, at the barrier. 



The magnitude of this added hole current is determined in the follow- 

 ing way. It is assumed that all holes which enter the barrier region are 

 drawn into the contact by the field existing there. The number of holes 



