Page 313 - Electrical Properties of Materials

Lasers

12

What goes up must come down.
It’s all done with mirrors.
19th century aphorisms

12.1 Equilibrium
We have several times arrived at useful results by using the concept of equilib-
rium. It is a pretty basic tenet of science and as with a similar idea, conservation
of energy, it is always coming in handy. When we say that the electrons in a
solid have a Fermi–Dirac distribution of energies, we are really saying two
things: first, that the system is in equilibrium; second, that it has a particular
temperature. Temperature is a statistical concept and is bound up with the idea
of equilibrium. On the one hand, we cannot meaningfully speak of the temper-
ature of a single particle; on the other, if we have a system of particles that is
perturbed from equilibrium, say by accelerating some of them, then for a tran-
sient period the temperature cannot be specified, since there is no value of T
that will make the Fermi function describe the actual distribution. Of course,
for electrons in a solid, or atoms in the gaseous state, the effect of collisions
rapidly flattens out the perturbation, the whole system returns to its equilib-
rium state, and the idea of temperature becomes valid again, although its actual
value may have changed.
We have on one or two occasions considered perturbed equilibrium. We
saw, for example, in Chapter 1 that large currents may flow in a conductor
with a very slight change in the energy distribution. Thus, we could describe
low field conduction in metals and semiconductors without departing from the
equilibrium picture.
Lasers are different. They have massively perturbed population distribu-
tions that are nevertheless in some kind of equilibrium. But when we come
to consider what temperature corresponds to that equilibrium, it turns out to
be negative. Now you know that 0 K is a temperature that can never quite be ∗ Erewhon (approximately ‘nowhere’
obtained by the most elaborate refrigerator; so how can we get a negative tem- backwards) was a country in the book
of thesamenamebySamuelButler,
perature? It is not inconsistent really because, as we shall show, a negative
where all habits and beliefs were the
temperature is hotter than the greatest positive temperature. But before going opposite of ours and were justified with
∗
further into Erewhon let us return to earth and start from the beginning. impeccable logic and reasonableness.

12.2 Two-state systems
Let us consider a material in which atoms have only two narrow allowed energy
levels, as illustrated in Fig. 12.1. Provided that the whole system containing
the material is in thermal equilibrium, the two allowed levels will be popu-
lated corresponding to a dynamic energy equilibrium between the atoms. The