696 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



from (3) we get the condition for large injection 



el » 0.07 amp/cm^ in Si (51) 



Since tliis current in (51) is much less than do , we may quite properly 

 speak of large injection n^ P and small currents / « !„ at the same 

 time. 



Let us denote by 



ICM = iP/Uif (Ins + Ips) (52) 



the current density at which conductivity modulation starts to be im- 

 portant. Then we may distinguish three ranges of current: (a) very small 

 current / < I cm for which large injection analysis does not apply; (b) 

 low current I cm < I < h for which large injection analysis applies, but 

 the voltage drop Vp in the middle region is negligible; (c) large current 

 I > lo for which Vp is sizable. The treatment of this section has covered 

 ranges (b) and (c) . Range (c) (as treated here) does not extend to infinity 

 but only up to current densities of the order 



L 



8 X 10* amp/cm'^ 



p 



so that the diffusion currents in the contacts may be treated as a small 

 injection. 



Small Injection, No Recomhinaiion 



In this section, we shall cover ranges (a) and (b) in current density. 

 We must go back to the basic equations, but we shall make use of two 

 facts that have come out of the large injection analysis: (a) jSFp is negli- 

 gible when / « /o ; (b) 7 = no/n^ ^ 1 which means n(x) and p(x) are 

 essentially constant in the middle region ^ x ^ w when I <^ lo . 

 When we set 



no = n^o, Po = Pw (53) 



equations (16) give us 



noo = npV^^"^""^ (54) 



Pww — Pn ^ 



Then (11) gives 



I = I„^ I^= (/„, + /,,) [/^'^o+^-Li] (55) 



Now Vo + Vw is the total applied bias when Vp can be neglected ; there- 



