Semiconductor Statistics

Abstract

In this chapter we present three basic statistics that are commonly used in the derivation of distribution functions for gas molecules, photons and phonons, electrons in a metal, and electrons and holes in a semiconductor. These basic statistics are needed to deal with the problems of interactions of a large number of particles in a solid. Since a great deal of physical insight can be obtained from statistical analysis of the particle distribution functions in a solid, it is appropriate for us to devote this chapter to finding the distribution functions associated with different statistical mechanics for particles such as gas molecules, photons, phonons, electrons, and holes. The three basic statistics that govern the distribution of particles in a solid are (1) Maxwell–Boltzmann (M-B) statistics, (2) Bose–Einstein (B-E) statistics, and (3) Fermi–Dirac (F-D) statistics. The M-B statistics are also known as the classical statistics, since they apply only to particles with weak interactions among themselves. In the M-B statistics, the number of particles allowed in each quantum state is not restricted by the Pauli exclusion principle. Particles such as gas molecules in an ideal gas system and electrons and holes in a dilute semiconductor are examples that obey the M-B statistics. The B-E and F-D statistics are known as quantum statistics because their distribution functions are derived based on quantum-mechanical principles. Particles that obey the B-E and F-D statistics in general have a much higher density and stronger interaction among themselves than the classical particles. Particles that obey the B-E statistics, such as photons and phonons, are called bosons, while particles that obey the F-D statistics, such as electrons and holes in a degenerate semiconductor or electrons in a metal, are known as fermions. The main difference between the F-D and the B-E statistics is that the occupation number in each quantum state for the fermions is restricted by the Pauli exclusion principle, while bosons are not subjected to the restriction of the exclusion principle. The Pauli exclusion principle states that no more than two particles with opposite spin degeneracy can occupy the same quantum state. Therefore, the total number of particles with the same spin should be equal to or less than the total number of quantum states available for occupancy in a solid.