Theory of the Flow of Electrons and Holes in Germanium and 

 Other Semiconductors 



By W. VAN ROOSBROECK 



{Manuscript Received Mar. 30, 1950) 



A theoretical analysis of the flow of added current carriers in homogeneous 

 semiconductors is given. The simplifying assumption is made at the outset that 

 trapping effects may be neglected, and the subsequent treatment is intended 

 particularly for application to germanium. In a general formulation, differential 

 equations and boundary-condition relationships in suitable reduced variables and 

 parameters are derived from fundamental equations which take into account 

 I the phenomena of drift, diffusion, and recombination. This formulation is special- 

 ; ized so as to apply to the steady state of constant total current in a single car- 

 tesian distance coordinate, and properties of solutions which give the electro- 

 static field and the concentrations and flow densities of the added carriers are 

 discussed. The ratio of hole to electron concentration at thermal equilibrium 

 occurs as parameter. General solutions are given analytically in closed form for 

 the intrinsic semiconductor, for which the ratio is unity, and for some limiting 

 cases as well. Families of numerically obtained solutions dependent on a parame- 

 ter proportional to total current are given for w-type germanium for the ratio 

 equal to zero. The solutions are utilized in a consideration of simple boundary- 

 value problems concerning a single plane source in an infinite filament. 



Table of Contents 



1. Introduction 560 



2. General Formulation 565 



2.1 Outline 565 



2.2 Fundamental equations for the flow of electrons and holes 566 



2.3 Reduction of the fundamental ecjuations to dimensionless form 571 



2.31 The general case 571 



2.32 The intrinsic semiconductor 577 



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

 and properties of their solutions 578 



3. Solutions for the Steady State 583 



3 . 1 The intrinsic semiconductor 584 



3.2 The extrinsic semiconductor: w-type germanium 586 



3.3 Detailed properties of the solutions 588 



3.31 The behavior for small concentrations 590 



3.32 The zero-current solutions and the behavior for large concentrations. . . . 593 



4. Solutions of Simple Boundary- Value Problems for a Single Source 594 



5. Appendix 599 



5 . 1 The concentrations of ionized donors and acceptors 599 



5 . 2 The carrier concentrations at thermal equilibrium 600 



5.3 Series solutions for the extrinsic semiconductor in the steady state 601 



5 . 4 Symbols for quantities 605 



1. Introduction 



TN A semiconductor there are current carriers of two types: electrons 

 -*- in the conduction band, and positive holes in the filled valence band; 

 and the increase of their concentrations in the volume of the semicon- 



560 



