HOLE CONCENTIL\TIOX AXD POINT CONTACTS 475 



rent-voltage characteristic, h{\\). It is found empirically" that as long 

 as Vc is not too large (a few tenths of a volt for a point contact on n- 

 type germanium), it is a good approximation to take: 



/.) {Vc) = Ic {exp(^eV,/kT) - 1), (10) 



where Ic is a constant for a given contact. Except for the factor l3, this 

 is of the form to be expected from the diode theory of rectification. The 

 empirical value of 13 is usually less than the theoretical value of unity 

 in actual contacts. 



If (10) is inserted into (8), the following equation is obtained for the 

 current when there is an added concentration of carriers, pa, in the in- 

 terior: 



/ = /, (exp(l3eVe/kr) - 1) - «/,«. (11) 



Setting 7 = and solving the resulting equation for the floating po- 

 tential, Vc = Vcf, we find: 



Vcf = {kT/e8) log [1 + aif.JIc)]. (12) 



The floating potential may be simply related to the conductance cor- 

 responding to small current flow. Using Eqs. (9) and (11), we find: 



G = (dh/dVc)v,^y,, = {0eIc/kT) exp (l3eVcf/kT). (13) 



Since the normal low-voltage conductance is just 



Go = ^elc/kT, (14) 



we have 



G = Go exp (j3eVcf/kT). (15) 



By using (12), G can be expressed in terms of pa- This relation is given 

 and compared with experiment in Section III. Equation (15) may be 

 solved for the floating potential: 



Vcf = {kT/ei3) log (G/Go). (16) 



It should be noted that (16) does not involve pa directly. Thus it is pos- 

 sible to determine Vcf from a measurement of the change in conductance 

 without direct knowledge of the added hole concentration. It holds for 

 large as well as small pa. 



The logarithmic relation (16) between floating potential and conduc- 

 tance has been demonstrated by an experiment of Pearson. Theexperi- 



' See H. C. Torrev and C. A. Whitmer, "Crystal Rectifiers", McGraw-Hill Company, 

 New York, N. Y., (1949), p. 372-377. 



