490 



BELL SYSTEM TECHNICAL JOURNAL 



curred. In the present calculation it is assumed that recombination occurs 

 at the surface, but not in the volume. This is a good approximation for a 

 point contact on germanium. It is further assumed that the hole concen- 

 tration is sufficiently small so that Laplace's equation (29) may be used. 



The model which we shall use is illustrated in Fig. 9. The contact is in 

 the form of a circular disk of radius p on the surface of the semiconductor. 

 Cylindrical coordinates, r, 6, z, are used, with the origin at the center of 

 the disk and the positive direction of the s-axis running into the semi- 

 conductor. We calculate the flow due to the added holes, and shall use 

 the symbol p without subscript to denote the added hole concentration. 



Fig. 9. — Coordinates used for calculation of hole flow to contact area in form of circu- 

 ■'ar disk. 



With recombination at the surface, it is necessary to have a gradient in 

 the interior which brings the holes to the surface. 



It is assumed that the rate of recombination at the surface is: 



sp — holes/cm"-^, 



(lA) 



where the factor 5 has the dimensions of a velocity and p is evaluated at 

 the surface z = 0. According to measurements of Suhl and Shockley, 5 is 

 about 1500 cm/sec for a germanium surface treated with the ordinary 

 etch. The current flowing to the surface is: 



{iXpkT/e){dp/dz)z^o holes/cm- 



(2A) 



The l)oundary condition for p at the surface z — Q outside of the contact 

 area is obtained by ccjuating (lA) and (2A). This gives: 



where 



dp/dz = \p at z = 0, r > p 

 X = se/upkT, 



(3A) 

 (4A) 



