What is electronic band structure?

I'm gonna do it!  I'm going to attempt the impossible: to explain electronic band structure for a lay audience.

Why should you care about electronic band structure?

Condensed matter physics is one of the largest fields of physics with some of the biggest practical applications.  Pretty much all of our electronics are built on it.  Now, this knowledge is neither necessary, nor helpful to our everyday use of electronics.  But doesn't it bother you that you don't know the first thing about it?  That is, you don't have any idea what electronic band structure is?

There is a reason you've never heard of it.  Unless you've taken courses in quantum mechanics, electronic band structure is a very inaccessible concept.  And yet, it's one of the most basic concepts in condensed matter physics.  So let's do this!

First, a familiar concept

E = 1/2 mv²

Does this look familiar?  This is the equation for E, the kinetic energy of an object, given v, its velocity.  We're going to analyze the heck out of this equation.  Look, a graph!

In the above graph, v is simply a number, positive or negative.  But in reality, v is a vector, containing components in x direction, y direction, and z direction.  If we were to include two directions, this is what the graph would look like:

But that's complicated, so we'll just consider v in a single direction.

But now we have to add in quantum mechanics.  Even if you understand nothing about quantum mechanics, you probably know it has to do with combining the concepts of particles and waves.  It turns out that the velocity of a particle is proportional to how much the wave goes up and down per unit length.  We usually call this quantity k, the wavenumber, which is defined as 2 pi times the number of cycles per meter.

E is proportional to v², which is proportional to k².  And so, if we were to graph E vs k, it would look like this:

Congratulations!  We've just constructed an electronic band structure!

Band structure in a crystal

The band structure we constructed is the band structure in a vacuum.  That is, if we have electrons in a vacuum, then each electron will fall somewhere along that line.  But most electrons are not wandering freely in vacuums, they're trapped in sold objects.  For simplicity's sake, I will only consider the simplest of solid objects, a crystal.  A crystal is a solid which has a repeating structure.

Now I'm going to wave my hands around wildly.  Woooo!  Math omitted!  A repeating crystal structure leads to a repeating band structure.  (Mind you, the crystal is repeats in space, while the band structure repeats in k.  k is measured in units of inverse meters, so it's more like the reciprocal of space.)

But if the band structure is just repeating itself, then we might as well keep only the first copy.  In other words, we'll limit k to the "first Brillouin Zone".

Okay, but we forgot something.  The electrons are attracted to the atomic nuclei, and repelled by each other.  This changes the energy of the electrons in ways that are difficult to calculate.  But qualitatively, the effect is most noticeable whenever those lines cross each other.  That's because when the lines cross, it's easy for electrons to exist in a superposition of those two lines (and that's all the explanation you'll get out of me).

The dashed lines represent the original band structure, and the solid lines are our correction.

Okay.  So far, pretty simple (I see people in the audience shaking their heads saying, "Um... not simple, no.").  But as we add more dimensions, we can get even stranger-looking band structures.  For example, this is the band structure of graphene:

(image credit)

The horizontal axes are k in the x and y directions.  The vertical axis is E.  The black hexagon represents the first Brillouin Zone.

My point in showing this is to demonstrate that the electronic band structure can be quite complicated, and look very different for different kinds of solids.

Discreteness

And now I'm going to connect the band structure with another concept which you might find familiar.  In an atom, electrons have discrete energy states.  It's almost as if electrons are only allowed to be in certain orbits around the nucleus.

(Image credit)

Of course, this is not an accurate picture of electrons (which are waves, not just particles), but it's still true that electrons have discrete energy levels.  This is true of the electronic band structure as well.  I drew a continuous line, but in reality it is a set of discrete points.

And so, electrons are only allowed to have certain values of E and k.

How many points are there?  Well, how many atoms are there in the crystal?  The answer: millions of billions of billions.  So I might as well draw the band structure as continuous.

And yet, the fact that E and k are discrete has an important consequence.  According to the Pauli Exclusion Principle, no two electrons* may have the same values of E and k.  And we also know that there are millions of billions of billions of electrons.  As a result, the electrons fill up all those energy levels, starting with the lowest energies, going upwards, until we run out of electrons.

*ignoring spin

Note that if we ignore k, we actually get bands of allowed energies.  And sometimes there are gaps between these bands, where no electrons are allowed to exist.  Physicists call these energy bands, and energy gaps.  It's not uncommon for electrons to exactly fill up an entire energy band, right up to the energy gap.

Conclusion

The electronic band structure is the set of allowed energies and k-values of electrons.  In a vacuum, E just has to be proportional to k² (due to kinetic energy), but in a solid object, we have to consider the energy of attraction to nuclei and repulsion from other electrons.  This results in distortions in the relationship between E and k, and may even create energy gaps.  Energy gaps are values of energy which are forbidden to electrons.

Since there are lots of electrons, and they obey the Pauli Exclusion Principle, they fill up the electronic band structure, from the lowest energies upwards.

Now that I've explained the electronic band structure, you may look back at my post on semiconductors, which may make a bit more sense.  I also hope to write a few more essays explaining other things that should now make sense.  In the meantime, are there any questions?

All images, except those credited, were created by me.  They may be used if they are attributed.  I think 11 images in one post is some kind of record for me.