FLOW OF ELECTRONS AND HOLES IN GERMANIUM 579 



lor the w-type semiconductor with Po = 0, and 

 (25) ^ = ^ + ^ (p _ i)R 



for the intrinsic semiconductor, with R given as (1 + aP) by (16); P has 

 the same meaning in both equations, the concentration unit being «o 

 for each case. With time variations excluded in this way, the parameter C 

 is a constant and the cHfferential equations apply to the steady state of 

 constant current. 



Since the equations involve only the single independent variable X 

 which does not appear explicitly, their orders may be reduced by one, in 

 accordance with a well-known transformation, which consists in intro- 

 ducing P as a new independent variable, and 



d d 



as new dependent variable : Noting that -yr; is equivalent to <j~, the dif- 



ferential equations become 



(27) ^ = 

 ^ ' dP 



c-'-^o P 



=; + - 



1 + b+lp 



R 



[1 + m 



x + '-^p 







[1 + 2P\G 



for the ;/-type semiconductor, and 



dG h + \ {P - l)R 



(28) 



dP lb 



for the intrinsic semiconductor. These are differential equations of the 

 first order. 



The solutions sought in the semi-infinite region, A' > 0, are those for 

 which G = for AP = 0, that is, those which pass through the (AP, G) — 

 origin, where AP, which denotes P — Po, equals P for the w-type semi- 

 conductor and P— 1 for the intrinsic semiconductor. This condition is that 

 the concentration gradient vanish with the concentration of added holes, 

 as it must for X infinite. It will be shown that the differential equations 

 possess singular points at the (AP, G)-origin, and the physical interpretation 

 of the solutions through these singular points will be examined. For this 

 purpose, consider equation (27) for the w-type semiconductor which, in 

 the neighborhood of the origin, assumes the approximate form, 



^ ^ dP P ^ G' 



