Some Results Concerning the Partial Differential Equations 

 Describing the Flow of Holes and Electrons in Semiconductors 



By R. C. Prim, III 



(Manuscript Received June 22, 1951) 



The subject equations are investigated with the aim of establishing some 

 general properties of the flow fields which they describe, including the existence 

 or non-existence of classes of exact solutions having certain formal properties. 

 The results include a number of geometric characteristics of the vector fields 

 involved, a suggestive reformulation of the partial differential equations re- 

 stricting carrier concentration and electrostatic potential, and several classes of 

 exact solutions involving arbitrary constants and/or functions. Of particular 

 interest is a family of solutions in closed form for the steady-state, no-recom- 

 bination case involving an arbitrary harmonic function in three dimensions. 



Table of Contents 



A. Introduction 1174 



B. Some Properties of the Current Density Vector Fields 1177 



C. Formulation of Partial Differential Equation System Restricting (P and V 1180 



D. The Recombination Rate Function (R 1182f 



E. Addition of Arbitrary Time Functions to V and 3C 1183 



F. Summary of Solutions for No Recombination or Time Variation 1183 



G. Solutions With V = V(t) 1185 



H. Solutions With (P = (P(t), N y^ 1187 



I. Solutions With (P = (P(t), N = 1188 



J. Solutions With 3C = SC(t), N ^0 1188 



K. Solutions With V = t)((P, t), grad (9^0 1189 



L. Solutions With V = vlh, t), (P = (?ih, t), gradCP j^ 0, div grad /f = 0, iV 5^ . 1191 



M. Solutions With V = V{h,t),(9 = (?{h, /), grad (P 9^ 0, div grad h = 0, N = . 1198 



N. Construction of Solutions from Orthogonal Harmonic Fields, N 9^ 1202 



O. Construction of Solutions from Orthogonal Harmonic Fields, iV = 1202 



P. Superposition of a Harmonic 5C Field, N 9^ 1203 



Q. A Partial Differential Equation in Terms of 3C Alone, N 9^ 1203 



R. Sample Application of the Results of Section L: Spherical Symmetry, iV 5«^ . . . 1205 



S. Sample Application of the Results of Section M: Spherical Symmetry, N = . . 1210 



T, Summary List of Symbols 1212 



U. References 1213 



A. Introduction 



THIS paper is concerned with the system of relations describing the flow 

 of holes and electrons in the interior of a homogeneous semiconductor 

 subject to the assumption of constant temperature, electrical neutrality, 

 and constant difference in concentrations of ionized donor and acceptor 

 centers. These relations are: 



div II. --,[« + If] 



(1) 



div||„ = el(R + ^^| (2) 



1174 



