592 



BELL SYSTEM TECHNICAL JOURNAL 



equation for P, which may be written as 



(48) P = P, exp {-x/st). 



For a large aiding field, s reduces to the velocity of drift under this 

 field while, for a large opposing field, the magnitude of 5 is approximately 

 Dp/nEaT. For zero field, 5 equals the diffusion velocity {Dp/rY, which is a 

 diffusion distance for a mean lifetime divided by the mean lifetime. This 

 diffusion velocity can be specified in terms of its field equivalent, or the 

 field which gives an equal drift velocity, and for germanium it is found that 

 the equivalent field is about 8 volt cm~^ for r equal to one microsecond 

 and about 2.5 volt cm~^ for r equal to 10 microseconds. 



For small concentrations of added holes in the intrinsic semiconductor, 

 or (P— 1) < < 1, equations (38) and (40) give the approximate solutions, 



(49) 



2b 



P-l = (P«-l)exp[± pi + "'(*+« 



] 



I'] 



c„ = 



h+ \\_ 



C 



2b 



b+ 1 



b+ 1 



the X-origin being selected arbitrarily at the point at which the relative 

 concentration is P" according to the approximation. It is evident from the 

 equation for Cp that, for (P— 1) small, the transport velocity is the drift 

 velocity under the applied field, which is the velocity of the holes norm- 

 ally present in the semiconductor. The differential transport velocity, ob- 

 tainable by differentiating the equation for Cp with respect to P and 

 using the differential equation (28), or by writing the exponent in the 

 equation for (P— 1) in the form given in (48), is, on the other hand, given by 



(50) 



26 



L(l + a){b+ 1)J 





Dr 



1 



2DpDn 



_1 -f aDp-\- Dn 



and is a diffusion velocity. This holds for holes added in any concentra- 

 tion if a = 0, or for constant mean lifetime, since the first of equations 

 (49) is then the general solution given in (39). 



The nature of the flow for small concentrations of added carriers in the 

 general case, which depends on the parameter Po , is illustrated qualita- 

 tively by the w-type and intrinsic cases considered, for which Po is re- 

 spectively zero and infinite. Solutions for the general case are easily 

 evaluated analytically from the linear differential equation which results 

 from (17) if P — Po << ^ + Po . It can be shown from the field-aiding 

 steady-state solution that the ratio of the differential transport velocity 

 to the velocity, proportional to C, of drift under the applied field is for 

 O >> (1 + 2Po)Mo equal to the quantity l/Mn. This result is consistent 



