14 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



ductivity from the total number of uncompensated impurities per square 

 centimeter of sheet and the approjiriate moliility weighted to account 

 for impurity scattering. 



4.0 CALCULATION OF DESIGN PARAMETERS 



To calculate the parameters which determine emitter efficiency, transit 

 time, and base resistance it is assumed that the distribution of uncom- 

 pensated impurities is given by 



N(x) = Nicrfc f - N-2erJc^ + Nz (4.1) 



where A^i and A^2 are the surface concentrations of the emitter and base 

 impurity diffusants respectively, Li and L^ are their respective diffusion 

 lengths, and Nz is the original doping of the semiconductor into which 

 the impurities are diffused. The impurity diffusion lengths are defined as 



Li = 2 V/M and L2 = 2 ^Ddo (4.2) 



where the D's are the respective diffusion coefficients and the f's are the 

 diffusion times. 



Equation (4.1) can be reduced to 



r(^) = Ti erfc I - Ta erfc X^ + 1 (4.3) 



where 



For cases of interest here, r(^) will be zero at two points, a and 13, 

 and will have one minimum at ^'. In the transistor structure the emitter 

 junction occurs at ^ = ^v and the collector junction occurs at ^ = (3. 

 Thus the base width is determined by 13 — a. The extent of aiding and 

 retarding fields in the base is determined by ^'. The integral of (4.3) 

 from to a, I\ , and from o to ^, I2 , are the integrals of interest in (3.2) 

 and thus determine emitter efficiency. In addition I2 is the integral from 

 which base resistance can be calculated. 



The calculations which follow apply only for values of ri/r2 and To 

 greater than ten. Some of the simplifying assumptions which are made 

 will not apply at lower values of these parameters where the distribution 

 of both diffusants as well as the background doping affect the structure 

 in all three regions of the device. 



