610 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



lithium. As must be the case from the above mentioned agreement, the 

 values of Do computed from them agree with the diffusion data of Fuller 

 and Severiens almost perfectly. This is a very quick and sensitive 

 method (also probably exceedingly accurate) for determining diffusivi- 

 ties. For example the work already completed, in effect, represents the 

 determination of diffusivities of the order of 10~^^ cm^/sec within a 

 matter of an hour, and, no doubt, smaller diffusivities could be deter- 

 mined by doping more heavily with acceptor. 



XIV. THE EFFECT OF ION PAIRING ON ENERGY LEVELS 



It was predicted in Section VIII that ion pairing would drive the 

 electronic energy states of donors and acceptors from the forbidden 

 energy region. In this section it will be demonstrated by low temperature 

 Hall effect measurements that the addition of lithium to gallium-doped 

 germanium does indeed result in the removal of states from the forbidden 

 gap rather than in the simple compensation which occurs when a non- 

 mobile donor such as antimony is added. 



At low temperatures where carrier concentration, p, is less than the 

 donor concentration, it can be expressed in the form^^ 



Na - N, 

 V = 



. {^_]^Y exp [-EJkT] (14.1) 



where Na and No are the concentrations of acceptor and donor states, 

 respectively, irip , the effective mass of free holes, h, Plank's constant, 

 and Ea the ionization energy of the acceptor. The values of nip and Ea 

 are known for the group III acceptors. 



Lithium was added to a specimen of germanium known to contain 

 1.0 X 10 cm~ gallium atoms and a negligible amount of ordinary 

 donors. Carrier concentrations for this specimen were determined from 

 Hall coefficient measurements. The logarithm of this concentration is 

 shown in Fig. 29 plotted against reciprocal temperature. The high 

 temperature limit of this plot fixes N a — Nd at 1.15 X 10 cm~^. 



At low temperatures the curve exhibits an extended linear portion to 

 which (14.1) should apply. Evaluating (14.1) with p = 4.0 X 10^^ cm"' at 

 1/T = 0.06 deg~' and A^^ - A^^ = 1.15 x lO'' cm~^ we find that Nd = 

 2.6 X 10'* cm"' and A^^ = 1.4 X lO'' cm~l 



Therefore, the density of apparent acceptor states has been decreased 

 by 1.0 X 10'® - 1.4 X 10'' = 8.6 X 10^^ cm"l The added concentration 

 of lithium was 1.0 X 10^^ cm"' - 1.15 X 10^^ cm"' = 8.85 X lO'^cm"', 

 almost identical with the loss in concentration of acceptor states. This im- 

 plies (as would be expected) that the lithium is almost totallj^ paired. 



