Subsections


10.6 Nearly-Free Electrons

The free-electron energy spectrum does not have bands. Bands only form when some of the forces that the ambient solid exerts on the electrons are included. In this section, some of the mechanics of that process will be explored. The only force considered will be one given by a periodic lattice potential. The discussion will still ignore true electron-electron interactions, time variations of the lattice potential, lattice defects, etcetera.

In addition, to simplify the mathematics it will be assumed that the lattice potential is weak. That makes the approach here diametrically opposite to the one followed in the discussion of the one-di­men­sion­al crystals. There the starting point was electrons tightly bound to widely spaced atoms; the atom energy levels then corresponded to infinitely concentrated bands that fanned out when the distance between the atoms was reduced. Here the starting idea is free electrons in closely packed crystals for which the bands are completely fanned out so that there are no band gaps left. But it will be seen that when a bit of nontrivial lattice potential is added, energy gaps will appear.

The analysis will again be based on the Floquet energy eigenfunctions for the electrons. As noted in the previous section, they correspond to periodic boundary conditions for periods $2\ell_x$, $2\ell_y$, and $2\ell_z$. In case that the energy eigenfunctions for confined electrons are desired, they can be obtained from the Bloch solutions to be derived in this section in the following way: Take a Bloch solution and flip it over around the $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 plane, i.e. replace $x$ by $\vphantom0\raisebox{1.5pt}{$-$}$$x$. Subtract that from the original solution, and you have a solution that is zero at $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. And because of periodicity and odd symmetry, it will also be zero at $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\ell_x$. Repeat these steps in the $y$ and $z$ directions. It will produce energy eigenfunctions for electrons confined to a box 0 $\raisebox{.3pt}{$<$}$ $x$ $\raisebox{.3pt}{$<$}$ $\ell_x$, 0 $\raisebox{.3pt}{$<$}$ $y$ $\raisebox{.3pt}{$<$}$ $\ell_y$, 0 $\raisebox{.3pt}{$<$}$ $z$ $\raisebox{.3pt}{$<$}$ $\ell_z$. This method works as long as the lattice potential has enough symmetry that it does not change during the flip operations.

Figure 10.20: The red dot shows the wavenumber vector of a sample free electron wave function. It is to be corrected for the lattice potential.
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The approach will be to start with the solutions for force-free electrons and see how they change if a small, but nonzero lattice potential is added to the motion. It will be a “nearly-free electron model.” Consider a sample Floquet wave number as shown by the red dot in the wave number space figure 10.20. If there is no lattice potential, the corresponding energy eigenfunction is the free-electron one,

\begin{displaymath}
\pp{{\vec k},0}//// = \frac{1}{\sqrt{8\ell_x\ell_y\ell_z}}
e^{{\rm i}(k_x x + k_y y + k_z z)}
\end{displaymath}

where the subscript zero merely indicates that the lattice potential is zero. (This section will use the extended zone scheme because it is mathematically easiest.) If there is a lattice potential, the eigenfunction will change into a Bloch one of the form

\begin{displaymath}
\pp{{\vec k}}//// = \pp{{\rm p},{\vec k}}//// e^{{\rm i}(k_x x + k_y y + k_z z)}
\end{displaymath}

where $\pp{{\rm {p}},{\vec k}}////$ is periodic on an atomic scale. If the lattice potential is weak, as assumed here,

\begin{displaymath}
\pp{{\rm p},{\vec k}}//// \approx \frac{1}{\sqrt{8\ell_x\ell_y\ell_z}}
\end{displaymath}

Also, the energy will be almost the free-electron one:

\begin{displaymath}
{\vphantom' E}^{\rm e}_{\vec k}\approx {\vphantom' E}^{\rm e}_{{\vec k},0} = \frac{\hbar^2}{2m_{\rm e}} k^2
\end{displaymath}

However, that is not good enough. The interest here is in the changes in the energy due to the lattice potential, even if they are weak. So the first thing will be to figure out these energy changes.


10.6.1 Energy changes due to a weak lattice potential

Finding the energy changes due to a small change in a Hamiltonian can be done by a mathematical technique called “perturbation theory.” A full description and derivation are in {A.37} and {D.80}. This subsection will simply state the needed results.

The effects of a small change in a Hamiltonian, here being the weak lattice potential, are given in terms of the so-called Hamiltonian perturbation coefficients defined as

\begin{displaymath}
H_{{\vec k}\underline{\vec k}} \equiv \langle\pp{{\vec k},0}////\vert V\pp{\underline{\vec k},0}////\rangle
\end{displaymath} (10.13)

where $V$ is the lattice potential, and the $\pp{{\vec k},0}////$ are the free-electron energy eigenfunctions.

In those terms, the energy of the eigenfunction $\psi_{\vec k}$ with Floquet wave number ${\vec k}$ is

\begin{displaymath}
{\vphantom' E}^{\rm e}_{\vec k}\approx {\vphantom' E}^{\rm...
...{\vec k},0}-{\vphantom' E}^{\rm e}_{{\vec k},0}}
+ \ldots %
\end{displaymath} (10.14)

Here ${\vphantom' E}^{\rm e}_{{\vec k},0}$ is the free-electron energy. The dots stand for contributions that can be ignored for sufficiently weak potentials.

The first correction to the free-electron energy is the Hamiltonian perturbation coefficient $H_{{\vec k}{\vec k}}$. However, by writing out the inner product, it is seen that this perturbation coefficient is just the average lattice potential. Such a constant energy change is of no particular physical interest; it can be eliminated by redefining the zero level of the potential energy.

Figure 10.21: The grid of nonzero Hamiltonian perturbation coefficients and the problem sphere in wave number space.
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That makes the sum in (10.14) the physically interesting change in energy. Now, unlike it seems from the given expression, it is not really necessary to sum over all free-electron energy eigenfunctions $\psi_{\underline{\vec k},0}$. The only Hamiltonian perturbation coefficients that are nonzero occur for the $\underline{\vec k}$ values shown in figure 10.21 as blue stars. They are spaced apart by amounts $J$ in each direction, where $J$ is the large number of physical lattice cells in that direction. These claims can be verified by writing the lattice potential as a Fourier series and then integrating the inner product. More elegantly, you can use the observation from addendum {A.37.3} that the only eigenfunctions that need to be considered are those with the same eigenvalues under displacement over the primitive vectors of the lattice. (Since the periodic lattice potential is the same after such displacements, these displacement operators commute with the Hamiltonian.)

The correct expression for the energy change has therefore now been identified. There is one caveat in the whole story, though. The above analysis is not justified if there are eigenfunctions $\pp{\underline{\vec k},0}////$ on the grid of blue stars that have the same free-electron energy ${\vphantom' E}^{\rm e}_{{\vec k},0}$ as the eigenfunction $\pp{{\vec k},0}////$. You can infer the problem from (10.14); you would be dividing by zero if that happened. You would have to fix the problem by using so-called “singular perturbation theory,” which is much more elaborate.

Fortunately, since the grid is so widely spaced, the problem occurs only for relatively few energy eigenfunctions $\pp{\vec k}////$. In particular, since the free-electron energy ${\vphantom' E}^{\rm e}_{{\vec k},0}$ equals $\hbar^2k^2$$\raisebox{.5pt}{$/$}$$2m_{\rm e}$, the square magnitude of $\underline{\vec k}$ would have to be the same as that of ${\vec k}$. In other words, $\underline{\vec k}$ would have to be on the same spherical surface around the origin as point ${\vec k}$. So, as long as the grid has no points other than ${\vec k}$ on the spherical surface, all is OK.


10.6.2 Discussion of the energy changes

The previous subsection determined how the energy changes from the free-electron gas values due to a small lattice potential. It was found that an energy level ${\vphantom' E}^{\rm e}_{{\vec k},0}$ without lattice potential changes due to the lattice potential by an amount:

\begin{displaymath}
\Delta{\vphantom' E}^{\rm e}_{{\vec k}} = - \sum_{\underli...
...underline{\vec k},0} - {\vphantom' E}^{\rm e}_{{\vec k},0}} %
\end{displaymath} (10.15)

where the $H_{{\vec k}\underline{\vec k}}$ were coefficients that depend on the details of the lattice potential; ${\vec k}$ was the wave number vector of the considered free-electron gas solution, shown as a red dot in the wavenumber space figure 10.21, $\underline{\vec k}$ was an summation index over the blue grid points of that figure, and ${\vphantom' E}^{\rm e}_{\underline{\vec k},0}$ and ${\vphantom' E}^{\rm e}_{{\vec k},0}$ were proportional to the square distances from the origin to points $\underline{\vec k}$, respectively ${\vec k}$. ${\vphantom' E}^{\rm e}_{{\vec k},0}$ is also the energy level of the eigenfunction without lattice potential.

The expression above for the energy change is not valid when ${\vphantom' E}^{\rm e}_{\underline{\vec k},0}$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\vphantom' E}^{\rm e}_{{\vec k},0}$, in which case it would incorrectly give infinite change in energy. However, it is does apply when ${\vphantom' E}^{\rm e}_{\underline{\vec k},0}$ $\vphantom0\raisebox{1.1pt}{$\approx$}$ ${\vphantom' E}^{\rm e}_{{\vec k},0}$, in which case it predicts unusually large changes in energy. The condition ${\vphantom' E}^{\rm e}_{\underline{\vec k},0}$ $\vphantom0\raisebox{1.1pt}{$\approx$}$ ${\vphantom' E}^{\rm e}_{{\vec k},0}$ means that a blue star $\underline{\vec k}$ on the grid in figure 10.21 is almost the same distance from the origin as the red point ${\vec k}$ itself.

Figure 10.22: Tearing apart of the wave number space energies.
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One case for which this happens is when the wave number vector ${\vec k}$ is right next to one of the boundaries of the Wigner-Seitz cell around the origin. Whenever a ${\vec k}$ is on the verge of leaving this cell, one of its lattice points is on the verge of getting in. As an example, figure 10.22 shows two neighboring states ${\vec k}$ straddling the right-hand vertical plane of the cell, as well as their lattice $\underline{\vec k}$ values that cause the unusually large energy changes.

For the left of the two states, ${\vphantom' E}^{\rm e}_{\underline{\vec k},0}$ is just a bit larger than ${\vphantom' E}^{\rm e}_{{\vec k}.0}$, so the energy change (10.15) due to the lattice potential is large and negative. All energy decreases will be represented graphically by moving the points towards the origin, in order that the distance from the origin continues to indicate the energy of the state. That means that the left state will move strongly towards the origin. Consider now the other state just to the right; ${\vphantom' E}^{\rm e}_{\underline{\vec k},0}$ for that state is just a bit less than ${\vphantom' E}^{\rm e}_{{\vec k},0}$, so the energy change of this state will be large and positive; graphically, this point will move strongly away from the origin. The result is that the energy levels are torn apart along the surface of the Wigner-Seitz cell.

Figure 10.23: Effect of a lattice potential on the energy. The energy is represented by the square distance from the origin, and is relative to the energy at the origin.
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That is illustrated for an arbitrarily chosen example lattice potential in figure 10.23. It is another reason why the Wigner-Seitz cell around the origin, i.e. the first Brillouin zone, is particularly important. For different lattices than the simple cubic one considered here, it is still the distance from the origin that is the deciding factor, so in general, it is the Wigner-Seitz cell, rather than some parallelepiped-shaped primitive cell along whose surfaces the energies get torn apart.

Figure 10.24: Bragg planes seen in wave number space cross section.
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But notice in figure 10.23 that the energy levels get torn apart along many more surfaces than just the surface of the first Brillouin zone. In general, it can be seen that tears occur in wave number space along all the perpendicular bisector planes, or Bragg planes, between the points of the reciprocal lattice and the origin. Figure 10.24 shows their intersections with the cross section $k_z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 as thin black lines. The $k_x$ and $k_y$ axes were left away to clarify that they do not hide any lines.

Recall that the Bragg planes are also the boundaries of the fragments that make up the various Brillouin zones. In fact the first Brillouin zone is the cube or Wigner-Seitz cell around the origin; (the square around the origin in the cross section figure 10.24). The second zone consists of six pyramid-shaped regions whose bases are the faces of the cube; (the four triangles sharing a side with the square in the cross section figure 10.24). They can be pushed into the first Brillouin zone using the fundamental translation vectors to combine into a Wigner-Seitz cell shape.

Figure 10.25: Occupied states for the energies of figure 10.23 if there are two valence electrons per lattice cell. Left: energy. Right: wave numbers.
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Figure 10.26: Smaller lattice potential. From top to bottom shows one, two and three valence electrons per lattice cell. Left: energy. Right: wave numbers.
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For a sufficiently strong lattice potential like the one in figure 10.23, the energy levels in the first Brillouin zone, the center patch, are everywhere lower than in the remaining areas. Electrons will then occupy these states first, and since there are $J\times{J}\times{J}$ spatial states in the zone, two valence electrons per physical lattice cell will just fill it, figure 10.25. That produces an insulator whose electrons are stuck in a filled valence band. The electrons must jump an finite energy gap to reach the outlying regions if they want to do anything nontrivial. Since no particular requirements were put onto the lattice potential, the forming of bands is self-evidently a very general process.

The wave number space in the right half of figure 10.25 also illustrates that a lattice potential can change the Floquet wave number vectors that get occupied. For the free-electron gas, the occupied states formed a spherical region in terms of the wave number vectors, as shown in the middle of figure 10.17, but here the occupied states have become a cube, the Wigner-Seitz cell around the origin. The Fermi surface seen in the extended zone scheme is now no longer a spherical surface, but consists of the six faces of this cell.

But do not take this example too literally: the small-​perturbation analysis is invalid for the strong potential required for an insulator, and the real picture would look quite different. In particular, the roll-over of the states at the edge of the first Brillouin zone in the energy plot is a clear indication that the accuracy is poor. The error in the perturbation analysis is the largest for states immediately next to the Bragg planes. The example is given just to illustrate that the nearly-free electron model can indeed describe band gaps if taken far enough.

The nearly-free electron model is more reasonable for the smaller lattice forces experienced by valence electrons in metals. For example, at reduced strength, the same potential as before produces figure 10.26. Now the electrons have no trouble finding states of slightly higher energy, as it should be for a metal. Note, incidentally, that the Fermi surfaces in the right-hand graphs seem to meet the Bragg planes much more normally than the spherical free-electron surface. That leads to smoothing out of the corners of the surface seen in the periodic zone scheme. For example, imagine the center zone of the one valence electron wave number space periodically continued.