10.5 Free-Electron Gas

Chapter 6 discussed the model of non­interacting electrons in a periodic box. This simple model, due to Sommerfeld, is a first starting point for much analysis of solids. It was used to provide explan­ations of such effects as the incom­pressi­bility of solids and liquids, and of electrical conduction. This section will use the model to explain some of the analytical methods that are used to analyze electrons in crystals. A free-electron gas is a model for electrons in a crystal when the physical effect of the crystal structure on the electrons is ignored. The assumption is that the crystal structure is still there, but that it does not actually do anything to the electrons.

The single-particle energy eigen­functions of a periodic box are given by

$$\pp{\vec k}/{\skew0\vec r}/// = \frac{1}{\sqrt{{\cal V}}... ... = \frac{1}{\sqrt{{\cal V}}} e^{{\rm i}(k_xx+ k_y y + k_z z)}$$ (10.11)

Here the wave numbers are related to the box dimensions as

$$k_x = n_x \frac{2\pi}{\ell_x} \qquad k_y = n_y \frac{2\pi}{\ell_y} \qquad k_z = n_z \frac{2\pi}{\ell_z}$$ (10.12)

where the quantum numbers $n_x$, $n_y$, and $n_z$ are integers. This section will use the wave number vector, rather than the quantum numbers, to indicate the individual eigen­functions.

Note that each of these eigen­functions can be regarded as a Bloch wave: the exponentials are the Floquet ones, and the periodic parts are trivial constants. The latter reflects the fact the periodic potential itself is trivially constant (zero) for a free-electron gas.

Of course, there is a spin-up version $\pp{\vec k}////{\downarrow}$ and a spin-down version $\pp{\vec k}////{\uparrow}$ of each eigen­function above. However, spin will not be much of an issue in the analysis here.

The Floquet exponentials have not been shifted to any first Brillouin zone. In fact, since the electrons experi­ence no forces, as far as they are concerned, there is no crystal structure, hence no Brillouin zones.

10.5.1 Lattice for the free electrons

As far as the mathematics of free electrons is concerned, the box in which they are confined may as well be empty. However,it is useful to put the results in context of a surrounding crystal lattice anyway. That will allow some of the basic concepts of the solid mechanics of crystals to be defined within a simple setting.

It will therefore be assumed that there is a crystal lattice, but that its potential is zero. So the lattice does not affect the motion of the electrons. An appro­priate choice for this lattice must now be made. The plan is to keep the same Floquet wave number vectors as for the free electrons in a rectangular periodic box. Those wave numbers form a rectangular grid in wave number space as shown in figure 6.17 of chapter 6.18. To preserve these wave numbers, it is best to figure out a suitable reciprocal lattice first.

To do so, compare the general expression for the Fourier ${\vec k}_{\vec{m}}$ values that make up the reciprocal lattice:

$${\vec k}_{\vec m} = m_1 \vec D_1 + m_2 \vec D_2 + m_3 \vec D_3$$

in which $m_1$, $m_2$, and $m_3$ are integers, with the Floquet ${\vec k}$ values,

$${\vec k}= \nu_1 \vec D_1 + \nu_2 \vec D_2 + \nu_3 \vec D_3$$

(compare section 10.3.10.) Now $\nu_1$ is of the form $\nu_1$ $\vphantom0\raisebox{1.5pt}{$=$}$ $j_1$$\raisebox{.5pt}{$/$}$$J_1$ where $j_1$ is an integer just like $m_1$ is an integer, and $J_1$ is the number of lattice cells in the direction of the first primitive vector. For a macroscopic crystal, $J_1$ will be a very large number, so the conclusion must be that the Floquet wave numbers are spaced much more closely together than the Fourier ones. And so they are in the other two directions.

Figure 10.16: Assumed simple cubic reciprocal lattice, shown as black dots, in cross-section. The boundaries of the surrounding primitive cells are shown as thin red lines.

In particular, if it is assumed that there are an equal number of cells in each primitive direction, $J_1$ $\vphantom0\raisebox{1.5pt}{$=$}$ $J_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $J_3$ $\vphantom0\raisebox{1.5pt}{$=$}$ $J$, then the Fourier wave numbers are spaced farther apart than the Floquet ones by a factor $J$ in each direction. Such a reciprocal lattice is shown as fat black dots in figure 10.16.

Note that in this section, the wave number space will be shown only in the $k_z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 cross-section. A full three-di­mensional space, like the one of figure 6.17, would get very messy when crystal structure effects are added.

A lattice like the one shown in figure 10.16 is called a “simple cubic lattice,” and it is the easiest lattice that you can define. The primitive vectors are ortho­normal, just a multiple of the Cartesian unit vectors ${\hat\imath}$, ${\hat\jmath}$, and ${\hat k}$. Each lattice point can be taken to be the center of a primitive cell that is a cube, and this cubic primitive cell just happens to be the Wigner-Seitz cell too.

It is of course not that strange that the simple cubic lattice would work here, because the assumed wave number vectors were derived for electrons in a rectangular periodic box.

How about the physical lattice? That is easy too. The simple cubic lattice is its own reciprocal. So the physical crystal too consists of cubic cells stacked together. (Atomic scale ones, of course, for a physical lattice.) In particular, the wave numbers as shown in figure 10.16 corre­spond to a crystal that is macroscopi­cally a cube with equal sides $2\ell$, and that on atomic scale consists of $J\times{J}\times{J}$ identical cubic cells of size $d$ $\vphantom0\raisebox{1.5pt}{$=$}$ $2\ell$$\raisebox{.5pt}{$/$}$$J$. Here $J$, the number of atom-scale cells in each direction, will be a very large number, so $d$ will be very small.

In ${\vec k}$-space, $J$ is the number of Floquet points in each direction within a unit cell. Figure 10.16 would corre­spond to a physical crystal that has only 40 atoms in each direction. A real crystal would have many thousands, and the Floquet points would be much more densely spaced than could be shown in a figure like figure 10.16.

It should be pointed out that the simple cubic lattice, while definitely simple, is not that important physi­cally unless you happen to be particularly inter­ested in polonium or compounds like cesium chloride or beta brass. But the mathematics is really no different for other crystal structures, just messier, so the simple cubic lattice makes a good example. Furthermore, many other lattices feature cubic unit cells, even if these cells are a bit larger than the primitive cell. That means that the assumption of a potential that has cubic periodicity on an atomic scale is quite widely applicable.

10.5.2 Occupied states and Brillouin zones

The previous subsection chose the reciprocal lattice in wave number space to be the simple cubic one. The next question is how the occupied states show up in it. As usual, it will be assumed that the crystal is in the ground state, corre­sponding to zero absolute temperature.

As shown in figure 6.17, in the ground state the energy levels occupied by electrons form a sphere in wave number space. The surface of the sphere is the Fermi surface. The corre­sponding single-electron energy is the Fermi energy.

Figure 10.17: Occupied states for one, two, and three free electrons per physical lattice cell.

Figure 10.17 shows the occupied states in $k_z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 cross section if there are one, two, and three valence electrons per physical lattice cell. (In other words, if there are $J^3$, 2$J^3$, and 3$J^3$ valence electrons.) For one valence electron per lattice cell, the spherical region of occupied states stays within the first Brillouin zone, i.e. the Wigner-Seitz cell around the origin, though just barely. There are $J^3$ spatial states in a Wigner-Seitz cell, the same number as the number of physical lattice cells, and each can hold two electrons, (one spin up and one spin down,) so half the states in the first Brillouin zone are filled. For two electrons per lattice cell, there are just as many occupied spatial states as there are states within the first Brillouin zone. But since in the ground state, the occupied free electron states form a spherical region, rather than a cubic one, the occupied states spill over into immediately adjacent Wigner-Seitz cells. For three valence electrons per lattice cell, the occupied states spill over into still more neighboring Wigner-Seitz cells. (It is hard to see, but the diameter of the occupied sphere is slightly larger than the diagonal of the Wigner-Seitz cell cross-section.)

However, these results may show up presented in a different way in literature. The reason is that a Bloch-wave represen­tation is not unique. In terms of Bloch waves, the free-electron exponential solutions as used here can be represented in the form

$$\pp{\vec k}//// = e^{{\rm i}{\vec k}\cdot{\skew0\vec r}} \pp{{\rm p},{\vec k}}////$$

where the atom-scale periodic part $\pp{{\rm {p}},{\vec k}}////$ of the solution is a trivial constant. In addition, the Floquet wave number ${\vec k}$ can be in any Wigner-Seitz cell, however far away from the origin. Such a description is called an “extended zone scheme”.

This free-electron way of thinking about the solutions is often not the best way to understand the physics. Seen within a single physical lattice cell, a solution with a Floquet wave number in a Wigner-Seitz cell far from the origin looks like an extremely rapidly varying exponential. However, all of that atom-scale physics is in the crystal-scale Floquet exponential; the lattice-cell scale part $\pp{{\rm {p}},{\vec k}}////$ is a trivial constant. It may be better to shift the Floquet wave number to the Wigner-Seitz cell around the origin, the first Brillouin zone. That will turn the crystal-scale Floquet exponential into one that varies relatively slowly over the physical lattice cell; the rapid variation will now be absorbed into the lattice-cell part $\pp{{\rm {p}},{\vec k}}////$. This idea is called the “reduced zone scheme.” As long as the Floquet wave number vector is shifted to the first Brillouin zone by whole amounts of the primitive vectors of the reciprocal lattice, $\pp{{\rm {p}},{\vec k}}////$ will remain an atom-scale-periodic function; it will just become non­trivial. This shifting of the Floquet wave numbers to the first Brillouin zone is illustrated in figures 10.18a and 10.18b. The figures are for the case of three valence electrons per lattice cell, but with a slightly increased radius of the sphere to avoid visual ambiguity.

Figure 10.18: Redefinition of the occupied wave number vectors into Brillouin zones.

Now each Floquet wave number vector in the first Brillouin zone does no longer corre­spond to just one spatial energy eigen­function like in the extended zone scheme. There will now be multiple spatial eigen­functions, distin­guished by different lattice-scale variations $\pp{{\rm {p}},{\vec k}}////$. Compare that with the earlier approxi­mation of one-di­mensional crystals as widely separated atoms. That was in terms of different atomic wave functions like the 2s and 2p ones, not a single one, that were modulated by Floquet exponentials that varied relatively slowly over an atomic cell. In other words, the reduced zone scheme is the natural one for widely spaced atoms: the lattice scale parts $\pp{{\rm {p}},{\vec k}}////$ corre­spond to the different atomic energy eigen­functions. And since they take care of the non­trivial variations within each lattice cell, the Floquet exponentials become slowly varying ones.

But you might rightly feel that the critical Fermi surface is messed up pretty badly in the reduced zone scheme figure 10.18b. That does not seem to be such a hot idea, since the electrons near the Fermi surface are critical for the properties of metals. However, the picture can now be taken apart again to produce separate Brillouin zones. There is a construction credited to Harrison that is illustrated in figure 10.18c. For points that are covered by at least one fragment of the original sphere, (which means all points, here,) the first covering is moved into the first Brillouin zone. For points that are covered by at least two fragments of the original sphere, the second covering is moved into the second Brillouin zone. And so on.

Figure 10.19: Second, third, and fourth Brillouin zones seen in the periodic zone scheme.

Remember that in say electrical conduction, the electrons change occupied states near the Fermi surfaces. To simplify talking about that, physicist like to extend the pictures of the Brillouin zones periodi­cally, as illustrated in figure 10.19. This is called the “periodic zone scheme.” In this scheme, the boundaries of the Wigner-Seitz cells, which are normally not Fermi surfaces, are no longer a distracting factor. It may be noted that a bit of a lattice potential will round off the sharp corners in figure 10.19, increasing the esthetics.