6.12 Confinement and the DOS

The motion of a single particle in a confining box was described in chapter 3.5.9. Nontrivial motion in a direction in which the box is sufficiently narrow can become impossible. This section looks at what happens to the density of states for such a box. The density of states gives the number of single-particle states per unit energy range. It is interesting for many reasons. For example, for systems of electrons the density of states at the Fermi energy determines how many electrons in the box pick up thermal energy if the temperature is raised above zero. It also determines how many electrons will be involved in electrical conduction if their energy is raised.

By definition, the density of states

where

For a box that is not confining, the density of states is proportional
to

This gives the number of states that have energies less than some value

So the density of states is proportional to

Confinement changes the spacing between the states. Consider first
the case that the box containing the particles is very narrow in the

Consider the variation in the density of states for energies starting from zero. As long as the energy is less than that of the smaller blue sphere in figure 6.12, there are no states at or below that energy, so there is no density of states either. However, when the energy becomes just a bit higher than that of the smaller blue sphere, the sphere gobbles up quite a lot of states compared to the small box volume. That causes the density of states to jump up. However, after that jump, the density of states does not continue grow like the unconfined case. The unconfined case keeps gobbling up more and more circles of states when the energy grows. The confined case remains limited to a single circle until the energy hits that of the larger blue sphere. At that point, the density of states jumps up again. Through jumps like that, the confined density of states eventually starts resembling the unconfined case when the energy levels get high enough.

As shown to the right in the figure, the density of states is
piecewise constant for a quantum well. To understand why, note that
the number of states on a circle is proportional to its square radius

The next case is that the box is very narrow in the

The final possibility is that the box holding the particles is very narrow in all three directions. This produces a quantum dot or artificial atom. Now each energy state is a separate point, figure 6.14. The density of states is now zero unless the energy sphere exactly hits one of the individual points, in which case the density of states is infinite. So, the density of states is a set of spikes. Mathematically, the contribution of each state to the density of states is a delta function located at that energy.

(It may be pointed out that very strictly speaking, every density of
states is a set of delta functions. After all, the individual states
always remain discrete points, however extremely densely spaced they
might be. Only if you average the delta functions over a small energy
range

Key Points

- If one or more dimensions of a box holding a system of particles becomes very small, confinement effects show up.

- In particular, the density of states shows a staging behavior that is typical for each reduced dimensionality.