578 



BELL SYSTEM TECHNICAL JOURNAL 



in (22) owes its special form simply to this circumstance, while that for 

 Cp applies also to the general case, and the differential equation in P is a 

 consequence of the equations in P and W from div C and div Cp . Or, 

 in more detailed terms, since the ohmic contribution to C,, must be pro- 

 portional to C, div Cp contains only the contribution due to diffusion. 

 This is evident from the relationship obtained from (22), 



{2i) 



v/p — 



1 



&+ iL 



c - 



2b 



b+ 1 



grad P 



from which it foUow^s also that, despite the dependence of the local field 

 on concentration gradient, the ohmic contribution to the hole flow density 

 is the flow density of holes normally present in the intrinsic semiconduc- 

 tor under the unmodulated applied field. 



The equations which have been given for one-dimensional flow in the 

 «-type semiconductor can readily be transformed, in the manner indi- 

 cated, into the corresponding equations for the intrinsic semiconductor. 



2.4 Differential equations in one dimension for the steady state of constant 

 current and properties of their solutions 



The steady state of constant current in one dimension will be con- 

 sidered explicitly for two limiting cases: the «-type semiconductor with 

 Po = 0, and the intrinsic semiconductor. These serve to illustrate and 

 delimit the qualitative features of the general case. Furthermore, the case 

 Po = frequently applies as a good approximation^®, as does the intrinsic 

 case, which is of particular interest not only in itself but also because the 

 extrinsic semiconductor exhibits intrinsic behavior for large concentra- 

 tions, and because moderate increases in temperature above room tem- 

 perature, such as joule heating may produce, suffice to bring high back 

 voltage germanium into the intrinsic range of conductivity-^. The tem- 

 perature dependence of Po and of other reduced quantities is evaluated 

 for germanium in the Appendix. 



The ordinary differential equations in the reduced hole concentration, 

 P, for the steady state in one dimension, which result from equations 

 (17) and (22) by equating the time derivatives to zero are as follows: 



(24) 



p 



l + '-^p 



1 + 2P 



R 



-'"' In »-typc germanium of resistivity al)out 5 oiim cm, for example, tlic electron con- 

 centration exceeds the equilibrium hole concentration bj- a factor of about 70. 



" Germanium which is substantially intrinsic at room temperature has been produced: 

 R. N. Hall, paper 15 of the Oak Ridge Meeting of the American Physical Society, March 

 18. 1950. 



