600 BELL SYSTEM TECHNICAL JOURNAL 



can certainly be neglected. For an «-type semiconductor, for example, the 

 term {A~ — Aq) in Poisson's equation may be suppressed. This procedure 

 is strictly consistent with the neglect of p^ and go , but undoubtedly holds 

 to an even better approximation. If D is the total donor concentration in 

 the «-type semiconductor, the concentration of ionized donors may be 

 considered to satisfy the equation, 



(71) ^= H(D- D+) - KD'-n, 



ot 



which applies to the homogeneous semiconductor, with H and K con- 

 stants which characterize, respectively, the rate of ionization of unionized 

 donors, and the rate of recombination of an ionized donor with an electron. 

 If, as a result of a small thermal ionization energy, most of the donors are 

 ionized, so that KD/H < < 1, the change in ionized-donor concentration 

 for the steady state is given by (71) as 



(72) u^ - Dt ^-^{n- Wo), 



which is small compared with the corresponding change in electron con- 

 centration. In other cases, the use of the general expression obtainable 

 from (71) for the steady-state concentration of ionized donors in terms of 

 the electron concentration, or the expression for the other limiting case of 

 relatively few ionized donors, might provide a more precise description 

 provided the conditions under which solutions are sought do not involve 

 unduly rapid changes with time. 



5.2 The carrier concentrations at thermal equilibrium 



The ratio of the thermal-equilibrium values of the hole and electron 

 concentrations may be evaluated for «-type germanium from^ 



(73) 



np = 3- 10^-i ^ exp I — I = «» 



n — p = ns ~ Wo = l/biJLepo = 2.40-10'Vpo, 



where the electron concentration excess «s corresponds to complete ioniza- 

 tion of the donors, and is approximately Wq at the highest temperature at 

 which Pq is still negligible, which may be taken as room temperature-''. 

 The resistivity po is that which determines Uq . Thus, 



(74) Po = h [Vl + ^ni/noT - 1] , 



" loc. cit. 

 ' loc. cit. 



