26 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



band go into unoccupied a-traps. Since, according to postulate II, the 

 number of unoccupied traps is nearly equal to the total number of 

 traps, and is thus independent of 8Vb and 8p, the reverse flow will be 

 practically equal to the thermal equilibrium value. If Spa is the re- 

 combination cross-section for holes, the net flow of holes to a-traps is 



Upa = SpaVpipsiriai — Psna)(holes/cm^ scc). (23) 



Here, Vp is the velocity factor which when multiplied by the concen- 

 tration gives the number of holes crossing a unit area from one direction: 



Vp = (kT/2Trmp)K (24) 



With use of the relations, Uaj/ria = risi/ns , Pbi/pb = Psi/ps , from 

 equations (17) and (18), which follow from the fact that the a-traps are 

 in thermal equilibrium with the conduction band and the 6-traps with 

 the valence band, (23) becomes 



Upa = SpaVp{na/ns){psinsi — PsUs). (25) 



This expression may be simplified further. The ratio 



ria ^ Ng exp [{E, - Ea - eVs)/kT] ^ Na ^ ( E, - Ea \ . 

 n. Nc exp [{E, - Ec - eVs)/kT] Nc ^""^ \ kT J ^ ^ 



is independent of Ef and Vb . The ratio may be evaluated for an intrinsic 

 specimen with Vb = 0, in which case ria = Uao and ris = Ui . Thus 



Ua/ris = Uao/Ui . (27) 



We also have from postulate IV, 



PsiUsi = piTii . (28) 



The equilibrium products are 



PsUs = pn = n]. (29) 



With use of (27), (28) and (29), (25) becomes 



Upa = SpaVp{nao/ni){pini — pn). (30) 



Similarly, it is found that net flow of electrons to 6-traps is equal to: 



Unb = Sr^n{pbQ/nx){pini - pu) . (31) 



The total rate of recombination is given by the sum of Unb and Upa 

 and is given by an expression of the form : 



U --U^-^Upa^ Cipiui - pn) = C(n + p)8p. (32) 



