THEORY OF THE SWEPT INTRINSIC STRUCTURE 1257 



Thus the equivalent length of current generation is 2L even though the 

 current is actually being generated in an effective length L. The reason 

 is that for an NIP structure the holes entering the right hand half of the 

 / region were generated in the left hand side. For an IP structure the 

 holes entering the space charge regions from the left were injected at the 

 external left hand contact to the / region. 



Applied Voltage 



In all cases the voltage can be found from the area under the E versus 

 X curve. In Figs. 2 to 4 the area under the curves gives the voltage ac- 

 curately; recombination becomes important only where the field is so 

 low as to have a negligible effect on the total voltage drop. 



Correction of the Cubic 



To conclude this section we consider the error introduced by using the 

 assumption / = asE. For simplicity take Ei as the unit field, 2Li as 

 the unit length and aEi as the unit current. Then the cubic becomes 

 E" — I IE = x" — A. The corresponding exact solution is E' — s = 

 x^ — A where the relation between s and E is given by equation (2.6) 

 which in dimensionless form is 



^'%-^B-I (3.8) 



where £ is of the order of 10~ . 



Let bE and bs represent the difference between the cubic and the cor- 

 rect solution at a giv^en x. Assume that bE and its second derivative are 

 small compared to E and its second derivative respectively. Then bs = 

 2EbE and on the correct solution sE - I = (2lf -f I/E)bE. So (3.8) 

 becomes 



bE ( £' \d'-E 



E \2E^ + // dx'- 



(3.9) 



To obtain a first approximation to bE/E we use the cubic to evaluate 

 d E/dx . It is convenient to express the results in terms of a dimension- 

 less variable z = E/I^'^, or if E and / are measured in conventional units 

 z = E{a/Eilf'\ Then (3.9) becomes 



bE __ 1 (L,£\-" ( z \- , ( X Y ^'(1 - z) (3 ^Q^ 



E 2\U / \z^ + hJ \2LJ {z^ + i)^ 

 iV'hen the lengths are in conventional units. 



