1044 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 



Similarly, the flow of carrier-pairs to the surface (2) becomes: 



U = 



(6) 



x-iX-ie^(l + X5) x-'e-" + xXe"''(l + X'^S) xe" 



wliich, for 6-^0, tends to the linear law U = sniS, where s, the surface 

 recombination velocity, is given by: 



s/{:VTnVTpf'' = NtSt 



where 



St ={\ + X'')ian<7,y''/2\ch(p + fnx) +ch(Y - (n\ - fnx)] (7) 

 The surface density Sg of trapped charge is given by: 



= Nd\ (8) 



2., 



where ft is the occupancy factor, given by (5). 



Now let us turn to the question of a distribution of surface traps 

 through the energy v. Suppose that the density of states having v lying 

 lietween v and v -\- dp is N(v) dv, expressed in units (ni£). Then the total 

 surface recombination velocity arising from all traps, and the total 

 trapped surface charge density, are given by : 



s/ivrnVrpY" = ni£ J St{p)N{v) dv (9) 



2, = \ Jt{v)N{v)dp (10) 



where St{v) and /«(i') are explicit functions of v, given by (5) and (7). 

 The limits of the integrals in (9) and (10) are the values of v correspond- 

 ing to the conduction and valence band edges; however, as we shall see, 

 it is often possible to replace these limits by ± «= . 



In summing up the contributions in the way represented l\v (9), we 

 ha\'e implicitly ignored the possibility of inter-trap transitions, suppos- 

 ing that the popidation of each trap depends only on the rates of ex- 

 change of charge with the conduction and valence bands, and is inde- 

 pendent of the population of any other trap of differing energy. 



What kind of function do we expect N{v) to be? Brattain and Bardeen' 

 postulated that N{v) was of the form of two delta-functions, correspond- 

 ing to discrete trapping levels high and low in the band. This assumption 

 is not cousislciit with the observed facts in ri'gard to field cITi^-l, surface 



