DISTRIBUTION AND CROSS-SECTIONS OF GERMANIUM SURFACES 1045 



photo-voltage, or surface recombination velocit}'. The general difficult}' 

 is that the obser\'ed cjuantities usually vary less rapidly with surface 

 potential than one would expect. It is possible to fit the field-effect obser- 

 vations of Brown and Montgomer}'" with a larger number of discrete 

 levels, but this would call for a "sharpening up" of the trapped charge 

 distribution as the temperature is lowered, and this appears to be con- 

 trary to what is observed.* It is always possible that the surface is patchy, 

 in w^hich case almost any variation with mean surface potential could be 

 explained. The simplest assumption, however, seems to be that N{v) 

 is a rather smoothly-varying function. All we need assume for the 

 moment is that it is everywhere finite, continuous and differentiable. 

 We may then differentiate equation (10) with respect to Y and 5 under 

 the integral sign, and get {d^s/dY)^ and (5Ss/55)f, the cjuantities for 

 which experimental measurements were reported in the previous paper :^ 



i-^ = [- 

 \dYji J 4 



N{p) ch 



ch\h{v -f Y) - \tn X] 



N{v){h{\-' + \)m{v - Y) -f i ^n X 



+ In x] + \{\~' - X)) civ 



4.ch\h{v +Y) ~\ (n X] 



(11) 

 (12) 



Notice that the expression in brackets in the numerator of (12) gener- 

 ally has the value X~ or —X, except near the point v = Y — fn\ — 2fnx- 

 This is indicative of the fact that, whatever the exact form of N(v), the 

 ratio of — (32s/35)y/(3Ss/(9F)5 tends to these limiting values (X^^ and 

 —X) for sufficientlj^ large negative and positive Y respectively. 



It may be verified from (7), (11) and (12) that {dXs/dY)^ , found from 

 the field effect experiment, depends only on N(v) ; (d'Zs 88) y , found from 

 the surface photo- voltage, depends on N{v) and x; while s, the surface 

 recombination velocity, depends in addition on the geometric mean 

 cross-section (anapY''. Both x and (a-„(7p) '"^ might themselves, of course, 

 be functions of p. Thus relations (7), (11) and (12) are integral eciuations, 

 from which the three unknown functions of v may in principle be de- 

 duced from the experimental results. (Equation 11 , in fact, may be solved 

 explicitly. P. A. Wolff'^ has shown, how^ever, that, to determine N{v) 

 unambiguously, it is necessary to know (52^/(9 F)j for all values of Y 

 in the range ± ^ .) 



The foregoing considerations apply to "small-signal" measurements. 



* There are some changes with temperature, but not what one would expect if 

 there were only discrete surface states. 



