THEORY OF THE SWEPT INTRINSIC STRUCTURE 1247 



In actual semiconducting materials, recombination is not direct. 

 Rather it occurs through a trap, or recombination center. The statistics 

 of indirect recombination has been treated by Shockley and Read^ for a 

 recombination center having an arbitrary energy level &i somewhere in 

 the energy gap. At any temperature the trap level can be expressed by 

 the values rii and pi which n and p would have if, at that temperature, 

 the Fermi level were at the trap level. Shockley and Read showed that, 

 at a given temperature, the lifetime for small disturbances in carrier 

 density is a maximum in intrinsic material. It drops to limiting values 

 T„o and Tpo in highly extrinsic n and p material, respectively. The formula 

 for gr — r in terms of n and p is 



g - r = — T— - — . , ^. . r (2.14) 



Tpo(n + ni) + Tnoip + Pi) 



For our purposes it is more convenient to define the hfetime r not by 

 '''(d ~ f) = — 6n « Wi , but rather as the proportionality factor in the 

 mass action law. Then r is not necessarily constant independent of carrier 

 density. From (2.12) and (2.14) 



Tpo(n + Wi) + T„o(p + P\) i^ . re. 



r = ~ (2.15) 



We shall be interested in the hfetime in the region where 7i and p are 

 equal to or less than n, . r decreases as n and p decrease; that is, t is less 

 in a swept region than in normal intrinsic material. Let r = Tj for 7i = 

 p = 7ii and T = To for n = p = 0. The total range of variation of r is by 

 a factor of 



II = 1 + ^^'(t'po + rpo) ,^ ^g. 



TO PlTnO + rilTpO 



Let the energy levels be measured relative to the intrinsic level, and 

 define a level 8o by 



\ TpO 



Then if &t = &o , niTpo = piTno • Now eq. (2.16) becomes 



So = kT\n .., 



TpO 



^^^sech(^i^) (2.17) 



Thus the variation in r increases as the ratio of Tno to Zpo deviates from 

 unity and as the trap level moves away from the level 8o . 



3 W. Shockley and W. T. Read, Jr., Phys. Rev., 87, p. 835, 1952. 



