452 BELL SYSTEM TECHNICAL JOURNAL 



were uniform and the applied field were zero. As the field is increased 

 the relative contribution to the sum current from the central hill square 

 becomes larger and larger, while that from the valley squares becomes 

 less and less. At very large fields the curve for any subsquare ap- 

 proaches a straight line whose slope is the Schottky slope; hence the 

 sum curve also approaches a straight line having this same slope. 

 The area which is now contributing most of the current is, however, 

 considerably less than the entire area. Roughly speaking, one might 

 say that at low fields something more than half the area is "effective" 

 in emitting electrons; as the field increases the "effective" area de- 

 creases; at large fields and high values of ju, 50 per cent of the total 

 current comes from about 5 per cent of the total surface ; one might say 

 that the "effective" area is approximately twice as large as this or 10 

 per cent. The values of the "effective" areas depend on the value of 

 /x: as M increases the "effective" area decreases. 



Such average curves as the one shown in Fig. 15 depend on three 

 variables, b, p. and T. This dependence is illustrated in Figs. 16, 17 and 

 18; in each case two of the variables are kept constant. Figure 16 

 shows log ijiao vs. V F for three values of h. This curve is similar to 

 Fig. 9 for a single circular patch. All the curves still approach a 

 Schottky line at high values of V ^ but because of the averaging 

 process they do not start out from a common value when F = and 

 the initial slope is not equal to the Schottky slope. In both figures 

 as the size of the patch decreases, the curves get less steep and the 

 place at which the curves bend over toward the upper Schottky line 

 moves to higher values of F. Note also that beyond this bend, the 

 curves are approximately straight but only approximately and that the 

 values are still somewhat below the theoretical Schottky line. 



This theoretical line has an intercept given by 



log isliao = log (1/18) [exp fielkT + exp — ixejkT 



+ 4(exp ixellkT + exp - ixejlkT) 



+ 4 (exp ixej^kT + exp - ^xejAkT)']. (73) 



This equation which defines is is based on a subdivision of the hill and 

 valley checker into 18 subcheckers. The exponentials contain the 

 value of /3 for each type of subchecker, in this case 1, h and \. If each 

 checker were divided into a larger number of subcheckers the number of 

 terms would be increased, but fortunately the value of log isliao would 

 not be greatly affected. This is particularly true as long as ne/kT is 

 less than 5. Since e/kT -^ 0.1 this means that values of log is/iao are 

 essentially correct for ^u less than 0.5 volt. W'e have plotted log is/i„o 



