628 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



Since the definite integral is a constant (E5) shows that S is a Hnear 

 function of s/t^ a fact mentioned in section XL 



In order to make use of the measured dependence of 2 on -sfi to 

 determine diffusivities, the functions y.{y) and A^d(v) must be specified. 

 For the latter we shall assume the Fick's law solution " 



Ar„ = AT^" erf v (E6) 



going with constant Z), and "N d = as a boundary condition at a; = 

 at the surface. (In section XI the limitations of this assumption in the 

 presence of ion pairing and diffusion potential are discussed.) The v 

 dependence of \x is more complicated. In general, we shall be concerned 

 with electrical measurements in two extreme cases. In the first case 

 ion pairing, under the condition of measurement, is everywhere com- 

 plete so that the local density of scatterers will be given by 



l^A - NM (E7) 



In the other case ion pairing will be entirely absent, so that the local 

 scatterer density, will be specified by 



N^ + NoM (E8) 



In all experiments A^^ will be only slightly greater than Nd so that it 

 may be replaced by this quantity. Doing this, and substituting (E6) 

 into (E8) and (E9) gives 



No' erfc V = N{v) (E9) 



for the scattering density in the ion pairing case, and 



Nn'a + erf v) = N(v) (ElO) 



for the no pairing case. 



Since almost all our experiments have been in germanium we now 

 specialize our attention to that substance. However, the procedure in- 

 voked below can be applied to silicon as well. 



The dependence of hole mobility, n, on scattering density, A^, for ger- 

 manium at room temperature is shown in Fig. 30 taken from Prince's 

 data.^^ The integral in (E5) assumes the form 



Nz," [ fxCNiv)) eric vdv. (Ell) 



Jo 



Choosing N{v) as either (E9) or (ElO) and using Fig. 30 together with a 

 tabic of error functions makes the numerical evaluation of (Ell) possible. 

 Since N(v) given by (E9) or (ElO) depends on No, so will the integral. 



