574 BELL SYSTEM TECHNICAL JOURNAL 



in which the use of (1 + IP) as dependent variable may be desirable. 

 This variable is equal to the concentration of carriers of both kindb di- 

 vided by the excess of electron concentration over hole concentration, 

 which is a constant. 



The expression in the equations which specifies the recombination 

 rate may be written more simply. Since the lifetime ratio Q is unity for 

 P = Po, 



(15) PQ- p,= {P-Po)R, 



where R, which will be called the recombination function, depends on P 

 and also equals unity for P = Po . The lifetime ratio and the recombina- 

 tion function which, of course, differ in general, both equal unity for the 

 case of constant mean lifetime. Recombination of holes and electrons at a 

 rate proportional to the product of their concentrations, called mass-action 

 recombination, and recombination characterized by a constant mean life- 

 time for holes are frequently of interest. For a combination of independent 

 mechanisms of both types, it is easily seen that 



(Q ^ r/r,, = 1 + (7 (/>-/>o)/«o = 1 + a (P - P„)/(l + Po), 



(16) < a = t/t, , < a < 1 



[P = 1 + ap/m = 1 + aP/(l + Po), 



where t,. is the mean lifetime for small concentrations associated with 

 mass-action recombination alone, so that a = for constant mean life- 

 time, and a = I for mass-action recombination. If both recombination 

 mechanisms are operative, that of mass-action recombination will, of 

 course, determine the mean lifetime where the concentration of added 

 carriers is sufficiently large. 



Recent experiments have shown that the mean lifetime for holes in 

 n-type germanium can be increased materially, to at least 100 micro- 

 seconds, by minimizing surface recombination through decreases in sur- 

 face-to-volume ratios.^ On the other hand, comparatively short mean 

 lifetimes, of the order of one microsecond, occur in p-type germanium 

 produced, for example, from n-type by nucleon bombardment. It should be 

 possible to determine in various cases which recombination law would 

 provide the better approximation by use of the technique of H. Suhl and 

 W. Shockley of hole injection in the presence of a magnetic field-"' or by 

 the j)hotoelectric technique of F. S. Goucher-^. 



' loc. cit. 



=«H. Suhl and VV. Shockley, Pkys. Rev. 75 (10), 1617-1618; 76 (1), 180 (1949). 

 " F. S. Gouchcr, paper I 1 1 of the Oak Ridge Meeting of the American Physical So- 

 ciety, March 18, 1950; P/tys. Rev. 78 (6), 816 (1950). 



