/' ;/ jr.WJ'lO.XS I.\ .S/:.\lU().\J)l i TORS 455 



since T,, aiul r„ arc both functions of .v. W'c shall obtain an approximate self- 

 consistent (lifTusion length by assuming that the holes diffuse, on the average, 

 to just such a depth, A,, , that in uniform material of the type found at 

 /.;, , their ditTusion length would also be L,, . At a depth L,, , the value of 

 // is <//.;, so that !)}■ (.■?.2), T;, is 1 raLj, = nl/gaL,, . Thus we write 



//p - Dt„ = Dnr'gal.,. (3.15) 



We can solve the equation (3.15) for /.;, and a similar one for A„ and insert 

 the results in equation (3.13). For small / this gives 



5v^ 7 = 7?: + {kT/qi;) = ^-^"- (\ + ^) In (.v„/7.„) 



It is seen that for g large, the second term, corresponding to the rectifying 

 resistance, becomes small. For this case, as discussed above, v^p = v'n and 

 the e.xact integral for i?o should be used and the junction will give poor 

 rectification. 



It is also instructive to consider La as a variable. Increasing /,„ corre- 

 sponds to making the transition from p to n more gradual. It is evident 

 that varying Z„ changes the two terms of (3.16) in opposite directions so 

 that there wdll be an intermediate value of L^ for which the resistance of 

 the junction is a minimum. As La approaches zero, however, the second 

 term should be modified: If we imagine that in the transition region the 

 concentration {Nd — N^ varies only over a finite range, bounded by fixed 

 values «n and pp in the n- and /^-regions, then it is clear that the limiting 

 values of Ly and Z„ should be given not by (3.15) but by -s/Drp and \/bDTn 

 where Tp and t„ are evaluated in the ^-region and /^-region. This leads to a 

 limiting value for Is, which is given in equation (4.11) of the following 

 section. In the range for which (3.16) applies, however, the interesting 

 result holds that widening the transition region initially decreases the re- 

 sistance by furnishing a larger volume in which holes and electrons may com- 

 bine or be generated. 



The condition that 8(fj dominate the resistance is that the second term 

 of (3.16) be much larger than the first. This leads to the inequality 



where we have neglected various factors involving b, which are nearly unity, 

 and ln(.Tb//,a) (which must be about 4 for Ge since the conductivity at .V(, 

 is about exp(4) times the intrinsic conductivity). The quantity 



A„ = {Dur'g)'" (3.18) 



