572 BELL SYSTEM TECHNICAL JOURNAL 



originally Hp , is denoted by m for simplicity. If b is the ratio of electron to 

 hole mobility, o-q is given in general by 



(9) ao = fJie(b)io + po) = Mo bfxe{ih - po), Mo = 1 + -^ Po, 







the symbol .l/n being introduced for brevity. If />o < < ^^o , Mo is unity 

 and cTo equals byicn^ . 



The independent dimensionless distance variables are .Y, Y and Z, 

 where the distance unit, L,, , is a diffusion length for a hole for the mean 

 lifetime, r, the diffusion constant for holes being Dp . This mean lifetime 

 is the unit for the independent dimensionless time variable, U. The hole 

 and electron concentrations are measured in units of the excess in concen- 

 tration of electrons over holes^'*, «o — po , the reduced variables being P 

 and X, respectively. The reduced total current C is total current density 

 measured in units of the current density /o which flows in the semicon- 

 ductor with no added carriers under the characteristic field Eo , which is 

 a field such that a carrier would expend the energy kT in drifting with it 

 through the distance Lp. A more illuminating alternative description is 

 that C is the ratio of the average drift velocity of holes under the applied 

 or asymptotic field, £„ , to the hole diffusion velocity (Dp/r)''. The field 

 E„ is that which produces the current density I in the semiconductor 

 with no added carriers. The corresponding reduced hole and electron flow 

 densities are Cp and C„ . The electrostatic field measured in units of Eq 

 is denoted by F, and W is the corresponding reduced electrostatic poten- 

 tial. The lifetime ratio (7 is a function of P which characterizes the re- 

 combination process. While it appears from experiment that the recom- 

 bination rate for holes depends on both physical and chemical properties 

 of the semiconductor, in a particular semiconductor at given temperature 

 it may be considered to depend on hole concentration alone. 



Representative values for germanium of units in terms of which the 

 dimensionless quantities are defined are as follows: The mean lifetime r 

 may be of the order of 10~^ sec. With a mobility for holes" of 1700 cm^ 

 volt"' sec~^ in germanium single crystals at 300 deg abs, the length Lp 

 is then about 2 -10"- cm; the characteristic field, Eo , 1.2 volt cm~^; and 

 the current density /o , 0.12 ampere cm~- for a resistivity of 10 ohm cm. 



With these definitions'", the fundamental equations for a region free 

 from external sources, neglecting changes in the concentrations of ionized 



^ loc. cit. 



''■' The excess in concentration of electrons over holes is of course equal to that of 

 ionize<l donors over ionized acceptors. 



'■'" The definitions ^iven ai)i)car best if there is a region in which F — Pu is small, with 

 P(, 9^ 0. Modified dellnitions of the reduced flow clensities, in which the conductivity 

 ffn is re|)laced hy the conductivity bjjie{no— po) due to the excess electrons alone, result in 

 criuations obtainable formally by setting Mo eriual to unity. 



