N.9 Explanation of the band gaps

Chapter 6.21 showed that the spectra of the electrons of
solids have band gaps;

energy ranges for which there
are no quantum states for the electrons. These band gaps were
qualitatively explained as the remnants of the discrete electron
energy states of the individual atoms. These discrete energy states
spread out when multiple atoms start interacting, but not necessarily
enough to completely remove the gaps.

However, if you start from the free-electron gas point of view, it is much less clear why and when addition of just a bit of crystal potential would suddenly pop up band gaps out of nothing. If you are curious, this note is for you.

To understand what is going on, the Kronig & Penney model will be
used. The crystal

is again taken to be one-dimensional.
The potential consists again of a sequence of straight dips, as was
shown in green in 6.22. The dips represent the attraction
of the atoms on the individual atomic electrons. However, to allow an
easier comparison with the free-electron gas solutions, this time the
dips will taken far less deep than before. Think of it as a model for
a metal, where the outer electrons are only very weakly bound to their
atomic cores.

For these shallower atomic

dips, and for a crystal
consisting of very many atoms,

the energy levels are
as shown to the left in figure N.1. Note that for the
higher energies, this is generally speaking very similar to the energy
levels for the free-electron spectrum shown to the left. That should
be expected; why would the shallow potential energy dips have much of
an effect when the kinetic energy of the electron considered is very
large? But even for high energy levels, there are still occasional
thin gaps. At these gaps, the electron velocity plunges to zero.
Why are these gaps there?

To qualitatively understand what is going on, from here on it will be
assumed that the periodic crystal

consists of just 12
atoms,

(rather than, say, a million). Mathematically,
after twelve atoms, the quantum wave function becomes the same as it
was initially and the solution repeats. You may think of the twelve
atoms as physically being arranged in a ring shape.

To make things easier to understand, it is also desirable to switch
from the complex Bloch wave

wave functions to the
equivalent real ones. These real wave functions may be found as the
real and imaginary parts of the Bloch waves. That is easiest for the
free-electron gas, where the Bloch waves are simply complex
exponentials; the Euler identity says

So for the free-electron gas, the real wave functions are

For the Kronig-Penney model, the real wave functions are more complex than simple sines and cosines, but can be found the same way.

Note that normally there are two different wave functions for each
value of the wave number

For the free-electron gas, that leaves as only ground state wave
function

For the Kronig-Penney case, the situation is a less simple. Consider
first the ground state, shown in the picture at the bottom of figure
N.2. (In figure N.2 the height of a wave
function picture illustrates the relative amount of energy of that
wave function. So the ground state picture is at the bottom.) The
square magnitude of the ground state wave function, shown as the black
line, is no longer a constant. It is higher than average at the dips
in the potential, at the atoms.

It is lower than
average in between atoms.

So the electron is somewhat
more likely to be found near an atom that attracts it than in between
atoms. The electron reduces its potential energy that way. But it
cannot do this without limit; if the electron is only found at the
atoms, the reduced uncertainty in position increases the kinetic
energy more than the potential energy is lowered. The best compromise
is given by the black line at the bottom of figure N.2.

To understand the energy states above the ground state, a key concept
of the general mathematical properties of real one-dimensional wave
functions is needed: *The more zero crossings in the wave
function, the higher the energy.*
Qualitatively, the reason is not that hard to understand. The more
zero crossings, the more wildly the wave function swings back and
forward between positive and negative values, raising the kinetic
energy. In figure N.2 the number of zero crossings is
listed as

Note further that only even numbers of zero crossings

Note next that in almost all cases, there are two different
wave functions of the same energy at a given number of zero
crossings. That is because if you have one wave function at a given

The only way this can possibly fail, and does, is if each atomic cell has the same number of zeros as its next neighbor. So every atomic cell must have the same number of zeros in it.

That, then, is why it is possible at all that there is only one wave
function in the ground state. In the ground state there are no zeros,
so every atomic cell has the same number, none. Indeed, looking
closer, in the ground state the wave function is identical in every
atomic cell. Mathematically, for

The next possibility that the shifted wave function does not give a
different one occurs when every atomic cell has one zero crossing.
For a crystal

of 12 atomic cells, that requires that
there are

But even if shifting the wave function does not give you a second one,
still there must be two different eigenfunctions for each even
number of zeros

For one, this explains why for the ground state where

But for the special case of

Similarly there will be a band gap at

One thing that may still be counter-intuitive is why the right-hand

To see why it is possible, look more closely at the

Also note that for

However, because of the modulation, the

Similarly for the

So the only finite energy gaps occur when the number of zeros is a whole multiple of the number of atoms. And the gap is between the two states with that number of zeros.

And between the states immediately above and below the gaps, the
energy difference is even smaller than elsewhere in the band. That
makes the electron velocity

Since the wave functions at the edges of the bands have zero propagation velocity, electrons in these states cannot move through the crystal. Now an implicit result of the analysis above is that for these states, a whole multiple of the Bloch wave length must equal double the atomic spacing. The Bloch exponential can change sign going from one atomic cell to the next, then return to the original sign at the next cell, but nothing more. If you train a beam of electrons with a wave length like that onto the crystal, the beam cannot propagate and will be totally reflected. That is in fact a key result of the Bragg reflection theory of wave mechanics, (10.16) in chapter 10.7.2. Thus Bragg theory can provide an intuitive justification for some of the features of the band structure.

If you want to see mathematically that the propagation velocity is indeed zero at the band gaps, and you know linear algebra, you can find the derivation in {D.84}. That also explains how the wave function figures in figure N.2 were made.