DISTRIBUTION AND CROSS-SECTIONS OF GERMANIUM SURFACES 1057 



Then the total additional change in (pp and <pn across the space-charge 

 t-egion, arising from the departure in uniformity in the carrier densities 

 t) and n, is : 



A,, = -£. / (i - 1) ,,. 



Mn J \n no/ 



(2) 



Suppose now that the true surface recombination rate is infinite, so 

 that the ciuasi-Fermi levels must coincide at the surface, and: 



(Pp -f A<Pp = <pn -+- A(pn (3) 



These equations, together with the known space-charge equations, 

 icomplete the problem. Notice first, from (2), that A<pp will be large only 

 if there is a region in which p is small (F ^ 1), while A^„ is large onl}^ 

 when, in some region, n is small (F <3C — 1). Introducing the cjuantity 5, 

 approximating for 8 small, equating Ip and /„ and setting the result eciual 

 to sriid, one finds: 



F « - 1 



{Dn/£)(\"' + X"'^')e^'' 



F » 1 (4) 



The coefficients {Dn/S) and (Dp/£) are of the order of 4 X 10' cm/sec. 

 The most extreme case encountered in our work is that occurring at the 

 ozone extreme for the n-type sample (X = 0.34, F = —6), for which the 

 surface recombination velocity, if limited by space-charge resistance 

 alone, would be about one-quarter of this (10'' cm/sec). The fact that 

 the observed surface recombination velocity is lower than that by more 

 than two orders of magnitude shows that space-charge resistance is not 

 a limiting factor in the present experiments. Equations 4 might Avell 

 hold on a sand-blasted surface, however, where the trap density is much 

 higher. 



'!-)* 



References 



1. W. L. Brown, Surface Potential and Surface Charge Distribution from Semi- 



conductor Field Effect Measurements, Phys. Rev. 98, p. 1565, June 1, 1955. 



2. W. H. Brattain and J. Bardeen, Surface Properties of Germanium, B.S.T.J., 



32, pp. 1-41, Jan., 1953. 



3. W. H. Brattain and C. G. B. Garrett, page 1019 of this issue. 



