PRJXCIPLES OF TRAXSISTOR ACTION 261 



cited electron and the hole left behind contribute to the conductivity. 

 The conductivities of all samples approach the same limitinj^ values re- 

 gardless of impurity concentration, given by an equation of the form 



CT = (^^exp {-Ea'2kT), (34) 



where k is Boltzmann's constant. P'or germanium, a^ is about 3.3 X 10'* 

 mhos cm and Ea about 0.75 ev. 



The exponential factor comes from the variation of concentration with 

 temperature. Statistical theory^-' indicates that ite and Uh depend on tem- 

 perature as 



n.= CeT'"exp(-^,/kT) (3.5a) 



nH = C,r''~expi-<p>,/kT) (3.5b) 



where ^pe is the energ>^ difference between the bottom of the conduction 

 band and the Fermi level and (ph is the difference between the Fermi level 

 and the top of the filled band. The position of the Fermi level depends on 

 the impurity concentration and on temperature. The theorv chives 



Ce^Ch^ 2{2Trmk/h-y'- ~ 5 X 10'^ (3.6) 



where m is an effective mass for the electrons (or holes) and h is Plank's 

 constant. The numerical value is obtained by using the ordinar>^ electron 

 mass for m. 



The product neUk is independent of the position of the Fermi level, and 

 thus of impurity concentration, and depends only on the temperature. 

 From Eqs. (3.5a) and (3.5b) 



iieiih = CeChT^ exp { — Ea/kT). (3.7) 



In the intrinsic range, we may set ;;, = n,, = ;;, and find, using (3.1), 

 (3.2), and {?>.i), an expression of the form (3.4) for a with 



<j^ = 11.5 X 10«aCX;,)^X (3.8) 



Using the theoretical value (3.6) for {CjChY'-, we find 



a^ = 0.9 X 10^ mhos/cm, 



as compared with the empirical value of ?>.3) X W, a difference of a factor 

 of 3.6. A similar discrepancy for silicon appears to be related to a varia- 

 tion of Ea with temperature. With an empirical value of 



CJOh = 25 X KP' X 3.6- - 3 X UF, (3.9) 



Eq. (3.7) gives 



Heiik ~ lO-'/cm*' (3.10) 



