Electrons and Holes in Semiconductors at Equilibrium

Abstract

Purpose of this chapter is providing the equilibrium expressions of the electron and hole concentrations in a semiconductor. For comparison, the cases of insulators and conductors are discussed qualitatively, and the concepts of conduction band, valence band, and generation of an electron-hole pair are introduced. The important issue of the temperature dependence of the concentrations is also discussed. Then, the general expressions of the concentrations in an intrinsic semiconductor are worked out, followed by an estimate of the Fermi level’s position. Next, the equilibrium expressions are worked out again, this time in the case where substitutional impurities of the donor or acceptor type are present within the semiconductor. The mechanism by which donor-type dopants provide electrons to the conduction band, and acceptor-type dopants provide holes to the valence band, is explained. An important outcome of the analysis is that the introduction of suitable dopants makes the concentration of majority carriers practically independent of temperature, at least in a range of temperatures of practical interest for the functioning of integrated circuits. The simplifications due to the complete-ionization and non-degeneracy conditions are illustrated, along with the compensation effect. Finally, the theory is extended to the case of a non-uniform doping distribution, where the concentrations must be calculated self-consistently with the electric potential by solving the Poisson equation. The last section illustrates the band-gap narrowing phenomenon. In the complements, after a brief description of the relative importance of germanium, silicon, and gallium arsenide in the semiconductor industry, a qualitative analysis of the impurity levels is carried out by an extension of the Kronig-Penney model, and the calculation of the position of the impurity levels with respect to the band edges is carried out.