p-n JUNCTIONS IN SEMICONDUCTORS 441 



The coefficient in (2.11) has the dimensions of (length)" leading us to 

 define a quantity 



Ld = y/KkT/Airqpi = y/KkT/STTifiii '< 



= 2.1 X 10~' cm for Si with k = 12.5,' «. = 2 X 10'° cm"' (2.12) 



= 6.8 X 10~' cm for Ge with k = 19,* Ui = 3 X lO" cm"' 



where the subscript D for Debye emphasizes the similarity of Ld to the char- 

 acteristic length in the Debye-Hiickel theory of strong electrolytes. The 

 meaning of the Debye length is apparent from the behavior of the solution 

 in a region where pa is constant, and u differs only slightly from the value 

 iio which gives neutrality, with p, sinh wo = Pd • Under these conditions, 



-1-2 = {L'd^ cosh tio){u — «c) (2.13) 



so that u — Hq varies as exp (± .vs/cosh u^i/Ld). In general, we shall be in- 

 terested in cases in which the deviation of ii from Hq decays to a small value 

 in one direction. It is evident that the distance required to reduce the devia- 

 tion to \/e is Z-d/ \/cosh wo • If only small variations in pd occur within a dis- 

 tance Lo/Vcosh^Mo J then the semiconductor will be substantially neutral. 

 However, if a large variation of pa occurs in this distance, a region of local 

 space charge will occur. These two cases are illustrated in connection with 

 the potential distribution in a p-n junction. 



2.2 Potential Distribution in the Transition Region^ 



We shall discuss the case shown in Fig. 1 for which the charge density 

 due to donors and acceptors is given by 



Nd - Na = ax (2.14) 



This relationship defines a characteristic length La given by 



La - Ui/a (2.15) 



If La » Ld , the condition of electrical neutrality is fulfilled (Appendix VII) 



and u satisfies the equation 



sinh u — Pd/pi — ax/2ni = x/2La 



6 J. F. Mullaney, P/iys. Rev. 66, 326 (1944). 



6H. B. Briggs and VV. H. Brattain, Fliys. Rev., 75, 1705 (1949). 



^ Potential distributions in rectifying junctions between semiconductors and metals 

 have been discussed by many authors, in particular N. F. Mott, Froc. Roy. Soc. 171A, 

 27 (1939) and W. Schottky Zc//i-./. Pliysi/c 113, 367 (1939) 118, 539 (1942) and elsewhere. 

 A summary in English of Schottky's papers is given by J. JotTe, Electrical Communications 

 22, 217 (1945). All such theories are in principle similar in involving the solution of equa- 

 tions like (2.11). See, for example, H. Y. Fan, Pliys. Rev. 62, 388 (1942). 



