FLOW OF ELECTRONS AND HOLES IN GERMANIUM 



573 



donors and acceptors and neglecting space charge-^, are given in reduced 

 form as follows: 



^^ = -[^»ModivC, + PQ - Pol 

 oU 



dN 

 dU 



= -[6ModivC„ -]- PQ - Pol 



(10) i ^^ = m ^^'^ - °"'^^ ''^ = " 6F0 ^ ^^^^ ^''^ + ^°^ ^^ 



Cn = ^ [-KV - grad N] = -^^ -V grad [-TI' + log X] 



{P - Po) - (N - No) = P - .¥ + 1 = 

 F = -grad W, 

 and the reduced form of equations (5) is 



(11) Cp - c„ = c. 



These reduced equations may be simplified and two differential equa- 

 tions in the dependent variables P and W written as follows: 



\-bM, div C„ = div P grad [W + log P] = [PQ - Po] + ~ 



oU 



(12) 



I 



div C = 0, 



C = — S erad 



W 



b - 1 



log I 



where S is the conductivity <x in reduced form 

 (13) 



2 ^ ^ = ^^ -^ P 



(70 ^'iVo + A 



Mo 



l + '^P 



An alternative formulation, due to R. C. Prim, which is obtained by evalu- 

 ating div [C„ ± b Cp], consists of the two equations, 



,-*— div (1 + 2P) grad W = - ,-^ div grad [W -(1 + 2P)] 

 0—1 -\- I 



(14) 



[PQ - Po] + 



dP 



^1 It may be desirable to take space charge into account in cases involving high fre- 

 quencies or high resistivities. Poisson's equation and equations (3) are in reduced form, 



9F 

 P - N + I = bMoT div F and Cp - C„ = C - T ^., where T = e/iircroT. 



The term containing T may often be omitted from one of these equations, depending on 

 the nature of the particular case considered. 



