DISTRIBUTION AND CROSS-SECTIONS OF GERMANIUM SURFACES 1017 



8.1 ohm-cm p-type: 



= 9.7c/; [0.31 (F - (n X) - 0.5] (14) 



for 2 > (F - (n X) > -4 



For values of (F — (n X) less than —4, it appears that Ss is changing 

 more rapidly with F than is indicated by (13) and (14). We shall comment 

 on this point later. Excluding this region, we note that in both cases the 

 variation with F is everywhere slow in comparison with e^, and proceed 

 on the assumption that N{v) is a function of v that varies everywhere 

 slowly in comparison with c" . Then (11) indicates that there is one fairly 

 sharp maximum in the integrand in the range ± « , occurring at that 

 value of V which coincides with the Fermi level: 



V ^ -F + (n X (15) 



The integral in (11) could be evaluated in series about this point 

 (method of steepest descents). The zero-order approximation is got by 

 replacing 



i sech' \h{v + F) - \(n X] by 6(i^ + F - Cn X). 



Later we shall proceed to an exact solution, and we shall find that this 

 delta-function approximation is not too bad. From (11) we now find: 



-f F - (ri X) dv = N{-Y + (n. X) (15) 



This mathematical procedure will be seen to be eciuivalent to identify- 

 ing {d'Ls/dY)i with the density of states at the point in the gap which 

 coincides with the Fermi-level at the surface. Using (13) and (14), one 

 gets: 



22.6 ohm-cm n-type: 



N{v) = 4.5 chiOMv + 0.8) (16) 



8.1 ohm-cm p-type: 



N{v) = 9.7 chiQMv + 0.5) (17) 



As we shall see in the next section, the exact solutions differ from (16) 

 and (17) only in the coefficients preceding the hyperbolic cosines. 



Turning to the surface photo-voltage measurements, we take (12) 

 and again replace 



I sech' [^{v + }') - \tn XJ by h{v + Y - In X) 



