1176 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



the ^'inverse method" which has proved very useful in the study of vari- 

 ous non-Hnear partial differential equation systems in mechanics. In the 

 inverse method, one proceeds by formal devices suggested by the equations 

 under study to try to find families of solutions to the equations which in- 

 volve arbitrary constants or, preferably, arbitrary functions. This is done 

 without reference to any preconceived boundary value problems. After a 

 pool of such families of solutions is available, it can be examined from the 

 point of view of finding boundary value problems of interest consistent 

 with any of the solutions in hand. The likelihood of finding solutions of 

 interest in this way is of course greatly enhanced when the solutions in- 

 volve arbitrary functions. Aside from providing solutions of some useful 

 boundary value problems, the solutions found by the inverse method 

 constitute a reference bank of non-trivial exact solutions against which 

 to check numerical methods and approximation schemes (based, for ex- 

 ample, on the assumption that a particular term can be neglected) for 

 solving problems of more immediate practical interest. 



J. Bardeen has demonstrated (in Reference 2) how the steady-state be- 

 havior of contact-semiconductor combinations can be explained on the 

 basis of the characteristics of (1) the flow field inside the semiconductor 

 and (2) those of the barrier layer at the contact. The present study is con- 

 cerned in this connection only with the first of these influences. It provides, 

 for example, a complete solution for the spherically symmetric flow field 

 without recombination for arbitrary currents^ — a generalization of the zero- 

 total current solution given by Bardeen. In the absence of surface recom- 

 bination this spherically symmetric solution provides the hemispherically 

 symmetric flow field in the neighborhood of a point contact on a plane 

 surface and remote from other electrodes or surfaces. This spherically sym- 

 metric solution is contained as a particular case in a family of solutions 

 involving an arbitrary harmonic function in three dimensions. Other choices 

 of the harmonic function can be made to yield flow fields associated with 

 numerous electrode configurations of immediate practical interest, for ex- 

 ample that of the type-A transistor. 



The objective of the present paper is to find (or establish the non-exist- 

 ence of) broad classes of solutions, and not to undertake detailed studies of 

 any particular solutions. Such detailed studies of particular cases from the 

 family of solutions mentioned above (and from other families found in 

 this study) will form the subject matter of papers dealing with specific flow 

 field configurations. However, in order to illustrate the interpretation of 

 mathematical arbitrary constants in terms of basic physical parameters, 

 the analysis of the spherically symmetric solution mentioned above is car- 



