438 BELL SYSTEM TECHNICAL JOURNAL 



Now from equation (50) 



dPo/dF\,^,^= dPo/dz\,^dZe/dF = FedZc/dF. (53) 



Hence from equations (52) and (53) 



d{^Pm)/dF = ezc. (54) 



Combining this with equation (47) we obtain 



d{Aip)/dF = Zc. (55) 



Now from 



log i = log U - 2 log T - (ip - A,p)e/2.3kT 



= log to + Aipe/2.?)kT (56) 

 we obtain 



d log i/dF = {d(A^)/dF)e/2.3kT. (57) 



Combining this with equation (55) we obtain 



d log i/dF = {e/2.3kT)zc. (58) 



This equation which was first derived by Becker and Mueller ^^ 

 allows us to obtain numerical values for Zc from the slope of the experi- 

 mental log i versus F curve. At Zc the surface field Fg is equal to the 

 applied field F. Hence a plot of Fs versus z can be obtained, and by 

 integrating this from s to cc , values of Pmo — Pq can be obtained for 

 various values of z greater than some minimum value corresponding 

 to the largest value of F. 



A particular case of a surface field, namely, that given by the image 

 law, is especially significant. In this case Fg = g/4s- and it can be 

 shown that the distances A and B in Fig. 7 are equal. At the critical 

 distance F = F^ and F = el\z^ or 



2. = {el\F)K (59) 



By substitution in equation (55) and integration from to F it follows 

 that 



A^ = ieF)K (60) 



Substituting this in equation (56) yields 



log i = log io + {ey2.3kT)ylF, (61) 



= log to + {\.9\/T)ylF. 



This equation, which was first derived by Schottky '^ and is called 

 the Schottky equation or law, predicts that a plot of log i versus ^F 

 should yield a straight line whose slope is e'^/2.3kT or 1.91/7". 



