272 BELL SYSTEM TECHNICAL JOURNAL 



If surface effects are neglected and it is assumed that holes flow radially 

 in all directions from the point contact, ^ may be expressed simply in terms 

 of the total hole current, //,, flowing from the contact: 



yP = -h/2Trr (4.28) 



Using (4.26), we may obtain the variation of hh with r. We are interested 

 in the limiting case in which w;, is'large compared with the normal electron 

 concentration, Uto. The logarithmic term in (4.26) can then be neglected, 

 and we have 



nh = Ik/AirrfXhkT. (4.29^ 



For example, if //, = 10~^ amps, nh = 10^ cmVvolt sec, and kT/e = .025 

 volts, we get, approximately, 



tih = 2X W/r. (4.30) 



For r '~ .0005 cm, the approximate radius of a point contact, 



nh ~ 4 X lOVcm^, (4.31) 



which is about 40 times the normal electron concentration in high-back- 

 voltage germanium. Thus the assumption that Uh is large compared with 

 Hco is valid, and remains valid up to a distance of the order of .005 cm, the 

 approximate distance the points are separated in the transistor. 

 To the same approximation, the field is 



F = kT/er, ' (4.32) 



independent of the magnitude of 7^. 



The voltage drop outside of the space-charge region can be obtained by 

 setting He in (4.19) equal to the value at the semi-conductor boundary of 

 the space-charge layer. This result holds generally, and does not depend 

 on the particular geometry we have assumed. It depends only on the 

 assumption that the electron current ie is everj^where zero. Actually u 

 will decrease and ie increase by recombination, and there will be an ad- 

 ditional spreading resistance for the electron current. 



If it is assumed that the concentration of holes at the metal-semi-conduc- 

 tor interface is independent of applied voltage and that the resistive drop 

 in the barrier layer itself is negligible, that part of the applied voltage which 

 appears across the barrier layer itself is: 



Vb = (kT/e) log in,i>/m,o), (4.33) 



where fihb is the hole concentration at the semi-conductor boundary of the 

 space charge layer and fiho is the normal concentration. For iihb -^ 5 X 

 10'« and ;/^o ~ 10'^, Vb is about 0.35 volts. 



