FLOW OF ELECTRONS AND HOLES IN GERMANIUM 



585 



provided a ± 0; for mass-action recombination a = 1. For a = or for 

 constant mean lifetime, (37) gives an exponential dependence of P— 1 on X: 



(39) 



F - i = {P' - 1) exp 



b + 1 " 



2b 



X 



where Po is the relative hole concentration for X = 0. Linear combina- 

 tions of the two solutions in (39) give solutions for composite cases, since 

 the differential equation from which (39) was derived is linear in P. A 

 similar result does not hold if there is mass-action recombination present, 

 and the more general procedure above referred to must then be followed. 



A characteristic feature of these solutions for the intrinsic semicon- 

 ductor is their independence of the current parameter C, this parameter 

 occurring only through a boundary condition, such as the one given in 

 equation (34) of Section 2.4. They are symmetrical in shape about a 

 source, the dependence of the concentration on the magnitude of the 

 distance from the source being the same for field opposing as for field 

 aiding, which follows quite simply from the symmetrical forms of the 

 solutions, and the condition that the concentration is everywhere con- 

 tinuous. 



Equations (22) and (23) of Section 2.32 provide the hole flow density 

 and the electrostatic field for this case. With G given for mass-action 

 recombination or for constant mean lifetime by the appropriate special 

 case of equation (36), and using the positive sign for an X-region to the 

 left of sources and the negative sign for an X-region to the right, 



(40) 



c - 



b - 



b-\- 



b-\- 



-A 



The electrostatic potential, V, is readily expressed in terms of F: From 



(41) 



F = 



eLp dV 



_e_dV 

 kTdX 



kT dP 



and (40), it is found that 



(42) 



whence 



_e_dV ^ b - 1 I _ r Jl. 

 kT dP b + I P GF' 



(43) ^ = ^^logP-c/ 



kT 



b+ 1 



dP_ 

 GP 



1 



b+ 1 



log P - C 



f dX 

 J P ' 



