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 ${\cal D}$ gives the number of single-particle states ${\rm d}{N}$ in an energy range from ${\vphantom' E}^{\rm p}$ to ${\vphantom' E}^{\rm p}+{\rm d}{\vphantom' E}^{\rm p}$ as

\begin{displaymath}
{\rm d}N = {\cal V}{\cal D}{\,\rm d}{\vphantom' E}^{\rm p}\qquad
\end{displaymath}

where ${\cal V}$ is the volume of the box containing the particles. To use this expression, the size of the energy range ${\rm d}{\vphantom' E}^{\rm p}$ should be small, but still big enough that the number of states ${\rm d}{N}$ in it remains large.

For a box that is not confining, the density of states is proportional to $\sqrt{{\vphantom' E}^{\rm p}}$. To understand why, consider first the total number of states $N$ that have energy less than some given value ${\vphantom' E}^{\rm p}$. For example, the wave number space to the left in figure 6.11 shows all states with energy less than the Fermi energy in red. Clearly, the number of such states is about proportional to the volume of the octant of the sphere that holds them. And that volume is in turn proportional to the cube of the sphere radius $k$, which is proportional to $\sqrt{{\vphantom' E}^{\rm p}}$, (6.4), so

\begin{displaymath}
N = \mbox{(some constant) } \left({\vphantom' E}^{\rm p}\right)^{3/2}
\end{displaymath}

This gives the number of states that have energies less than some value ${\vphantom' E}^{\rm p}$. To get the number of states in an energy range from ${\vphantom' E}^{\rm p}$ to ${\vphantom' E}^{\rm p}+{\rm d}{\vphantom' E}^{\rm p}$, take a differential:

\begin{displaymath}
{\rm d}N = \mbox{(some other constant) } \sqrt{{\vphantom' E}^{\rm p}} {\,\rm d}{\vphantom' E}^{\rm p}
\end{displaymath}

So the density of states is proportional to $\sqrt{{\vphantom' E}^{\rm p}}$. (The constant of proportionality is worked out in derivation {D.26}.) This density of states is shown as the width of the energy spectrum to the right in figure 6.11.

Figure 6.12: Severe confinement in the $y$-direction, as in a quantum well.
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Confinement changes the spacing between the states. Consider first the case that the box containing the particles is very narrow in the $y$-​direction only. That produces a quantum well, in which motion in the $y$-​direction is inhibited. In wave number space the states become spaced very far apart in the $k_y$-​direction. That is illustrated to the left in figure 6.12. The red states are again the ones with an energy below some given example value ${\vphantom' E}^{\rm p}$, say the Fermi energy. Clearly, now the number of states inside the red sphere is proportional not to its volume, but to the area of the quarter circle holding the red states. The density of states changes correspondingly, as shown to the right in figure 6.12.

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 $k_x^2+k_z^2$. That is the same as $k^2-k_y^2$, and $k^2$ is directly proportional to the energy ${\vphantom' E}^{\rm p}$. So the number of states varies linearly with energy, making its derivative, the density of states, constant. (The detailed mathematical expressions for the density of states for this case and the ones below can again be found in derivation {D.26}.)

Figure 6.13: Severe confinement in both the $y$ and $z$ directions, as in a quantum wire.
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The next case is that the box is very narrow in the $z$-​direction as well as in the $y$-​direction. This produces a quantum wire, where there is full freedom of motion only in the $x$-​direction. This case is shown in figure 6.13. Now the states separate into individual lines of states. The smaller blue sphere just reaches the line of states closest to the origin. There are no energy states until the energy exceeds the level of this blue sphere. Just above that level, a lot of states are encountered relative to the very small box volume, and the density of states jumps way up. When the energy increases further, however, the density of states comes down again: compared to the less confined cases, no new lines of states are added until the energy hits the level of the larger blue sphere. When the latter happens, the density of states jumps way up once again. Mathematically, the density of states produced by each line is proportional to the reciprocal square root of the excess energy above the one needed to reach the line.

Figure 6.14: Severe confinement in all three directions, as in a quantum dot or artificial atom.
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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 ${\rm d}{\vphantom' E}^{\rm p}$ do you get the smooth mathematical functions of the quantum wire, quantum well, and unconfined box. It is no big deal, as a perfect confining box does not exist anyway. In real life, energy spikes do broaden out bit; there is always some uncertainty in energy due to various effects.)


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

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