DISTRIBUTION AND CROSS-SECTIONS OF GERMANIUM SURFACES 1043 



>h , pi — values which the equilibrium electron and hole densities 

 at the surface would have if the Fermi level coincided with 

 the trapping level 

 Cn = NtVTnCTn ', Cp = NiVrpCTp , where iV^ stands for density of traps per 

 unit area, Vm is the thermal speed for electrons and Vtp that 

 for holes, and a„ and a,, are the cross-sections for transitions 

 between the traps and the conduction and valence bands 

 respectively. 

 If we introduce the surface potential Y and the c^uantity 5, defined as 

 (Ap/'Hi), where Ap is the added carrier density in the body of the semi- 

 conductor, we may write: 



ris = X~^/iie^(l + X5) 



Ps = Xn;e~^(l + \~^8) 



where X = po/ni , po being the e(iuilibriiun hole concentration in the body 

 of the semiconductor. We further introduce the notation: 



7ii = iiier" pi = nj-e" 



(4) 



(Cp/CnY = X 



The quantity v thus represents the energy difference, measured in 

 units of (kT/e), between the trapping level and the centre of the gap;* 

 and is positive for states below, negative for those above, this le^'el. The 

 parameter x ^vill be most directly associated m ith whether the state is 

 donor-like or acceptor-like. If it is donor-like (neutral or positive), a 

 transition involving an electron in the conduction band will be aided by 

 Coulomb attraction whereas one involving a hole will not; so one would 

 expect X « 1- For an acceptor-like trap, (neutral or negative) the con- 

 trary holds, and one expects x ^ 1- 



Using (4), the occupancy factor (1) becomes 



. ^ X~'X-Va + X3) + xe' 



' X-'\-'e^l + X5) + x-'e-" + xXe-'Xl + ^''8) + xe" (5) 



= iX~*e~*''e*'' sech ii {Y + v) - h (n X] for 5 = 



Note that, in thermodynamic ec[uilibrium, the occupancy factor does 

 not depend in any way on the cross-sections, whereas for 5 5^ it does, 

 through the ratio x- 



* Strictly speaking, one should say "position of the Fermi level for intrinsic 

 semiconductor" instead of "centre of the gap." These will fail to coincide if 

 the effective masses of holes and electrons are unequal, as they certainly are in 

 germanium. 



