FLOW OF ELECTRONS AND HOLES IN GERMANIUM 



577 



equations (8) to the desired new set, in which the ratio b of electron to 

 hole mobility is replaced by its reciprocal. 



2.32 The Inlrinsic semiconductor. 



For the intrinsic semiconductor, in which />o = Wo , the reduced concen- 

 trations given in (8) are inapplicable. As Pn approaches «o , these reduced 

 concentrations increase indefinitely, and the equations which those given 

 for the «-type semiconductor approach in the limit are homogeneous in 

 the concentration unit. These limiting equations therefore apply to the 

 intrinsic semiconductor in terms of a concentration unit which may be 

 chosen arbitrarily. The quantity ;?o will be chosen as this unit. Thus, 

 redefining the reduced concentration variables as 



X = n/uo ; 



N, 



(21) P ^ p/uo , 



from equations (12) and (14) any two of the equations in the dependent 

 variables P and W given by 



-{b -\- 1) div Cp = -^ div P grad W 



(22) { 



^^^ div grad P=[P()-1]+|^, 



C, = -__^-^Pgrad[ir+ logP]; 



div C - 0, 



C = -P grad 



W 



b+ 1 



logP 



and including the right-hand member which is common at least once, char- 

 acterize the intrinsic semiconductor'-^. 



It is noteworthy that one of these equations contains only P as de- 

 pendent variable, W being absent; and this equation indicates that the 

 spatial distribution of carrier concentration is not subject to drift under 

 the field, but only to a diffusion mechanism with diffusion constant 

 2DpD„/{Dp + Dn), where D,, = bDp is the diffusion constant for elec- 

 trons.-^ This result is readily accounted for as being due to a conductiv- 

 ity in the intrinsic case which is everywhere proportional to the concen- 

 tration of carriers of either type, so that 3 = P. The expression for C 



^^ These equations for the intrinsic case were first derived quite unambiguously as 

 those for the special case of the parameter po/n,, equal to unity in the general equations 

 written in terms of the concentration unit Ho. This unit is, however, less advantageous 

 than («(, — po) which, in obviating much of the formal dependence on po, makes for 

 greater generality. 



^^ The equations for the intrinsic case might be written in somewhat simpler form b}- 

 redefining the length unit in terms of 2D,,D,J(Dp -{- Dn) as a diffusion constant instead of 

 D,, , but their relationshi]i to those of the general case would then l)e less evident. 



