Page 173 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Statistics
states. This leads to the famous particle distributions. Armed with these
we can describe the behavior of many-particle systems in a very compact
manner. The rest of the chapter shows how this is done for bosons
(named after Bose, and include phonons and photons), and for fermions
(named after Enrico Fermi, and include electrons and holes).

5.1 Systems and Ensembles

In representing a system which is composed of a very large number N
A
of identical subsystems, there is a trade-off between the available
resources (i.e., computer memory, computation time) and the accuracy
N
that we can achieve for the representation. If is so large that the repre-
sentation of the exact state, including all subsystems, by far exceeds all
A
available resources, then the only way to represent the state of is by
means of statistical statements, i.e., probability distributions for the
respective system properties. In this chapter we develop the basic statisti-
cal techniques required to represent the ‘particle’ systems of our solid
state semiconductor. A thorough treatment of the basics may be found in
textbooks [5.1] and [5.2]. The application to semiconductors is dealt with
in [5.3].

Our discussion starts with the microcanonical ensemble, which considers
a large number of identical isolated systems. Each system resides in the
smallest possible energy range that we can possibly consider. This sys-
tem delivers us with a definition for the density of states. The microca-
nonical ensemble is extended in range to the next level, the canonical
ensemble, where we only allow heat exchange with a very large reser-
voir. This model delivers the canonical partition function. Stepping up,
we consider the grand ensemble, where we allow particle exchange with
the environment. From this model we obtain a definition for the chemical
potential, the driving force for particle exchange, as well as for the grand
partition function. With these results, we move on to statistically describe
particle counts. We specialize the statistics for different types of particle

170 Semiconductors for Micro and Nanosystem Technology