HOLE CONCENTRATION AND POINT CONTACTS 479 



with values of /3 obtained directly from the current-voltage characteristic 

 of the probe. Such a comparison would provide a valuable test of the 

 theory. 



III. Low Voltage Conductance of Point Contacts 



In this section we calculate the hole current flowing in the body of the 

 germanium from diffusion and find an expression relating change of con- 

 ductance with added hole concentration. The results shall be applied to 

 data of Pearson and of Suhl. ^^'e need to derive Eq. (4) which gives the 

 hole current in terms of the added hole concentrations, pba, measured just 

 outside the barrier layer, and pa, measured deep in the interior. 



The model which is used for the calculation is illustrated in ¥\g. 1. 

 The diffusion equation for hole flow is to be solved subject to the bound- 

 ary conditions that p = pb just outside the barrier layer and p = pi Sit 

 large distances from the contact in the interior. It is assumed that the 

 total current flow is zero or small. 



We shall first derive the more general equations* which include flow 

 by the electric field as well as by diffusion in order to show the conditions 

 under which the electric field can be neglected. In the body of the semi- 

 conductor, conditions of electric neutrality require that the electron con- 

 centration, n, be given by: 



n = Nf + p, (19) 



where Xf, the net concentration of fixed charge, is the difference between 

 the concentrations of donor and acceptor ions. We shall assume that 

 Nf is constant so that 



grad n = grad p. (20) 



The general equations for electron and hole current densities, /„ and ip, 

 are: 



in = M'l (f»^ -\- kT grad n) (21) 



ij, = Mp (epF - kT grad p), (22) 



where F is the electric field strength. By using (19) and (20), and setting 

 Hn = l^fJ'p, we can express /„ in the form: 



/„ = Vp ie{.\r + p) F -f kT grad p). (23) 



The magnitude of F for zero net current, 



i = /■;, + in = 0, (24) 



•* A discussion of the equations of flow is given in the article by VV. van Roosbroeck in 

 this issue of the Bell System Technical Journal. 



