584 



BELL SYSTEM TECHNICAL JOURNAL 



flow density Cp , and the reduced electrostatic field F, in terms of reduced 

 distance .Y are found in general by numerical means, which include nu- 

 merical integration and the evaluation of appropriate series expansions. 



General solutions which have been evaluated numerically for w-type 

 germanium for a number of values of the current parameter are given in 

 the figures. In the limiting cases of P small and P large, analytical ap- 

 proximations for the extrinsic semiconductor are readily obtained, that 

 for P large being derived from an analytical solution for C equal to zero, 

 or zero current. If the steady-state problem for the extrinsic semicon- 

 ductor is simplified by neglecting either recombination or diffusion, solu- 

 tions are obtainable which, like the zero-current one, are expressible in 

 closed form. 



For the intrinsic semiconductor, the general problem considered in this 

 section is solved quite simply by analytical means. The solution provides, 

 as physical considerations indicate it should, the same analytical approxi- 

 mation for large P as does the zero-current solution for the extrinsic case. 

 It may be well to consider first the intrinsic semiconductor which, aside 

 from the extrinsic semiconductor for the case of zero current, appears to 

 constitute the only analytically solvable steady-state case in one dimension 

 which has physical generality according to the present approach. 



3 . 1 The intrinsic semiconduclor 



Integrating the differential equation (28), it is found that 

 ^2 ^ + 1 



(35) 



/(. 



\)RdP, 



with R given as 1 -f ai' by (16), for an arbitrary combination of the two 

 recombination mechanisms, assumed independent. Thus 



(36) 



o' = ^4i(. 



1)'' 



(1 + a) + - a{P 



1) 



for the cases of field opposing or field aiding, for which G = for P— 1 = 

 0; for a composite case, a suitable constant is included on the right-hand 

 side. Excluding composite cases, the root may be taken in (36) and G 

 replaced by its definition, which gives 



(37) 



d{P - 1) 

 dX 



= ± 



[P - 1] 



(1 + a) -f ^ 



a{P - 1)1 



and if the .Y-origin is selected more or less arbitrarily as the point at 

 which P is infinite, then (37) gives 



"(1 + a){b + 1)7 ^ 



(38) 



1 = ^(i+^ csch^ 

 2a 



Sb 



