p-n JUNCTIONS IN SEMICONDUCTORS 471 



where, by I ^o {>p), we mean the hole current which would flow in the forward 

 direction across either /i or J2 if uninfluenced by the other (i.e. the function 

 of (4.11) or (4.18) and (4.20).) The equation shows that a fraction csch 

 {2ic/L,^ of the current Ipo (<^«), which would be injected by ^, on Pi in the 

 absence of Jy , flows into Pn . The conductance of P2 across Jo is increased 

 by the factor coth(2te'/Z,p). 



The current into Po carried by electrons wiU be unaffected by Ji and can 

 be denoted by -Ino{<Pc) the minus sign resulting from the fact that currents 

 into P2 are in the reverse direction. The total current flowing into P2 contains 

 the -/„o(<^c) and -Ipoi^c) terms and must cancel the +/j,o((pe) term for 

 equilibrium. Hence: 



Inoiipc) + coth (2w/Lj,) Ipo(<Pc) = csch {2w/Lp) Ij,oi<P*) (5.7) 



If pn » lip , the Ino term can be neglected compared to coth {2w/Lp) Ipo . 

 Hence the value of (pc must satisfy 



lA<Pc) = sech {2w/Lp) Ipo(<P,). (5.8) 



For <p, > kT/q, the exponential approximation may be used for Ipo in both 

 terms: 



^c = ^.- (kT/q) In cosh {2w/Lp), (5.9) 



so that, if (2w/Lp) is the order of unity, <pc should be only about (kT/q) 

 less than (^, . For (2u>/Lp) large, we get 



^. = <P,- (kT/q) {2w/Lp) (5.10) 



corresponding to the Imear drop of v^p , discussed in connection with equation 

 (4.9), across the distance 2w. 



WTien ^pe is negative, so that we have to deal with reverse current, ^c 

 win not decrease indefinitely but will reach a minimum value given by 



[exp qiPc/kT\ - 1 - -sech {2w/Lp) (5.11) 



and corresponding to saturation reverse current across /i , so that 



^, = -{kT/q) In [1 + (1/2) c^ch\w/Lp)]. (5.12) 



The floating potentials of /»-type contacts to «-type material into which 

 holes have been injected (or >i-type contacts to /'-type material with in- 

 jected electrons) are reminiscent of probes in gas discharges which tend to 

 become charged negative in respect to the space around them because they 

 catch electrons more easily than positive ions. The situation may also be 

 compared with that producing thermal e.m.f.'s; in fact a "concentration 

 temperature" of the semiconductor with injected holes can be defined by 

 finding the temperature for which np = n\{T). We conclude that, in the 



