24 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



We shall first relate Vd and Vb . Since the electric field in the film is 

 ^Trai/Ko , 



Vn = 4.TtDcri/Kn , (11) 



where Kd is the dielectric constant of the film. Substituting for ai 

 from Equation (3) and making use of (8) and (9), we find 



Vd = 2H sinh ^{Vb - Vbo) + const, (12) 



where 



H = ^ {NaN, exp [(E, - Ea)/kT])^ = (47reVi^x>)(n„oP6o)^ (13) 

 Kd 



If Vd is expressed in volts and tn in cm. 



i/ = 1.8 X 10-\£n/Kn)(naoPbQf. (13a) 



The contact potential Equation (2) may be expressed in the form 



c.p. = Vb - Vbo + 2H sinh ^(Vb - Vbo) + const. (14) 



The change in contact potential with illumination results from a 

 potential drop 8Vi in the interior and a drop 8Vb across the space- 

 charge layer. The former comes from the recombination current of 

 holes and electrons diffusing from the surface to the interior. Since 

 electrical neutrality requires that 8n = 8p, the concentration gradients 

 are equal. However, the mobility of electrons is greater than that of 

 holes, so that the diffusion current of electrons is larger than that of 

 holes by the mobility ratio "6." Since there can be no net current flow 

 to the interior, an electric field is established which is in such a direction 

 as to enhance the flow of holes and retard the flow of electrons. The net 

 potential drop associated with this electric field is 



/3(6 4-1) i bn+ p ) 



where 8p = bn is the change in concentration in the interior just beyond 

 the space-charge layer.* 



This potential is positive for both n- and p-type material, and is 

 independent of ambient, since 8p is independent of ambient. The ob- 

 served (Ac.p.)i, is generally much larger than given by (15) and is of 

 opposite sign for n- and p-type material. To account for the observa- 

 tions, it is necessary to assume that the major part of the effect is 

 associated with a space-charge layer at the free surface. We believe that 

 the present experiments give the most convincing evidence obtained so 

 far for the existence of' such a space-charge layer. ^ 



