260 BELL SYSTEM TECHNICAL JOURNAL 



being required. Both Ed and Ea are so small in germanium that practi- 

 cally all donors and acceptors are ionized at room temperature. If only 

 donors are present, the concentration of conduction electrons is equal to 

 the concentration of donors, and the conductivity is n-type. If only ac- 

 ceptors are present, the concentration of missing electrons, or holes, is 

 equal to that of the acceptors, and the conductivity is p-type. 



It is possible to have both donor and acceptor type impurities present in 

 the same crystal. In this case, electrons will be transferred from the donor 

 levels to the lower lying acceptor levels. The conductivity t\T3e then 

 depends on which is in excess, and the concentration of carriers is equal to 

 the difference between the concentrations of donors and acceptors. It is 

 probable that impurities of both types are present in high-back-voltage 

 germanium. The relative numbers in solid solution can be changed by 

 heat treatment, thus changing the conductivity and even the conductivity 

 type.^^ The material used in rectifiers and transistors has a concentration 

 of conduction electrons of the order of lO^Vc.c, which is about one for 

 each 5 X 10^ atoms. 



The conductivity depends on the concentrations and mobilities of the 

 carriers: Let /x^ and jXh be the mobilities, expressed in crnVvolt sec, and We 

 and fih the concentrations (number/cm^) of the electrons and holes respec- 

 tively. If both t^'pes of carriers are present, the conductivity, in mhos/cm, 

 is 



a = iieCfie + nhenh, (3.1) 



where e is the electronic charge in coulombs (1.6 X 10~^'). 



Except for relatively high concentrations ('~ lO^Vcni^ or larger), or at 

 low temperatures, the mobilities in germanium are determined mainly by 

 lattice scattering and so should be approximately the same in different 

 samples. Approximate values, estimated from Hall and resistivity data 

 obtained at Purdue University^^ and at the Bell Laboratories^^ are: 



MA - 5 X 10«r-3''2, (3.2) 



M. = 7.5 X 10«r-='/2 (cmVvolt sec), (3.3) 



in which T is the absolute temperature. There is a considerable spread 

 among the different measurements, possibly arising from inhomogeneity 

 of the samples. The temperature variation is as indicated by theory. 

 These equations give ma '^ 1000 and Mc ^^ 1500 cmVvolt sec at room tem- 

 perature. The resistivity of high-back-voltage germanium varies from 

 about 1 to 30 ohm cm, corresponding to values of ih between 1.5 X 10'^ 

 and 4 X lO'Vcm'. 



At high temperatures electrons may be thermally excited from the tilled 

 band to the conduction band, an energy Eo being required. Both the ex- 



