Pictures of Band Theory: A real space view of where bands and band gaps come from

In learning solid state physics, one of the most difficult conceptual hurdles to overcome is to understand band theory. This is partly due to the difficulty in thinking about reciprocal space, and is highlighted on Nanoscale Views blog in the post “The Tyranny of Reciprocal Space”. In this post, I will sacrifice accuracy in favor of an intuitive picture of band theory in real space. Hopefully, this post will help newcomers overcome those scary feelings when first exposed to solid state physics.

Firstly, it is necessary to recount the mathematical form of a Bloch wavefunction:

\psi_{k}(r) = e^{ikr}u(r)

Let’s pause for a second to take a look at what this means — the Bloch wave consists of a plane wave portion multiplied by a periodic function. In this post, for illustration purposes, I’ll simplify this by treating both parts of the Bloch wave as real.1 Take a look  at the image below to see what this implies:

image117

Fig 1: (a) The periodic potential. (b) The Bloch wavefunction. (c) The periodic part of the Bloch wave function. (d) The sinusoidal envelope part of the Bloch wavefunction.

Within this seemingly simple picture, one can explain the origin of band structure and why band gaps appear.

Let’s see first how band structure arises. For ease, since most readers of this blog are likely familiar with the solution to the infinite square well problem, we shall start there. Pictured below is a periodic potential with infinitely high walls between each well and the first two wavefunctions for each well looks like so:

PeriodicInfiniteSquareWell

Fig. 2: n=1 and n=2 wavefunctions for the periodic infinite square well.

The wavefunctions from well to well don’t have to be in phase, but I’ve just drawn them that way for ease. Bands arise when we reduce the height between walls to let the wavefunctions bleed over into the neighboring wells. This most easily seen for the two-well potential case as seen below:

 

 

In the first row, I have just plotted the n=1 energy levels for each well. Once the barrier height has been reduced, the (formerly degenerate) energy levels split into a symmetric and anti-symmetric state. I have not plotted the n=2 levels — this is just what happens if the n=1 interact! How much the energy levels split will be determined by how much I reduce the barrier height: the more I reduce the barrier, the larger the splitting. In band language, as you’ll see below, this implies that the lower the barrier height, the greater the dispersion.

One important thing to take away from this picture is that both in the infinite and finite barrier cases, we can fit at most four electrons in these two levels (if we include spin). In the infinite barrier case, two electrons can fit in the n=1 level in each well, and in the finite barrier case, two electrons can go into the symmetric state and two in the anti-symmetric state.

Now, let’s return to the case where we have an infinite  number (okay, I only drew fifteen!) of finite potential wells. In analogy to the two-well problem, we can draw the states for the case where the heights of the potential wells have been reduced:

PeriodicFiniteSquareWell

Fig. 3: n=1 and n=2 wavefunctions for the periodic finite square well. My lack of artistic skills is severely exposed for the n=2 level here, but imagine that the wavefunctions don’t look so discontinuous.

 

This is where things get interesting. How do we represent the n=1 states in analogy with the symmetric and anti-symmetric states in the two-well case? We can invoke Bloch’s theorem. It basically says that you just multiply this periodic part by a sinusoidal function!

The sinusoidal function ends up being an envelope function, just like in the very first figure above. Here is what the lowest energy level would look like for the periodic finite potential well:

PeriodicFiniteSquareWellBloch

Fig. 4: The lowest energy wavefunction for the n=1 level

This state is the analog of the symmetric state in the two-well case. To preserve the number of states in going from the infinite barrier case to the finite barrier case, I can only multiply the periodic part by N sinusoidal envelope functions, where N is the number of potential wells — in this case, fifteen!

Therefore the functions from the n=1 level end up looking like this:

 

PeriodicFiniteSquareWellBlochLevels2

Fig. 5: Wavefunctions that comprise the n=1 band

 

These are the wavefunctions that comprise a single band, that is, the band formed by the n=1 level. Interestingly, just from looking at the wavefunctions, you can see that the wavefunctions for the n=1 band increase in energy in going from the totally symmetric state to the totally antisymmetric state, as the number of nodes in the wavefunction increases. Notice here also how this connects to the reciprocal space picture — the totally antisymmetric wavefunction was multiplied with an envelope function that had wavelength 2a, which is the state at the Brillioun zone boundary!

Now, in this picture, why do band gaps exist? Understanding this point requires me to do the same envelope multiplication procedure to the n=2 levels. In particular, when one multiplies by the 2a envelope function, it essentially has the effect of flipping the wavefunction in each well so that we get something that looks something like this (again, imagine a continuous function here, my artistic skills fail me):

PeriodicFiniteSquareWellBloch2ndLevel

Fig. 6: The zone boundary (\pi/a) wavefunction for the n=2 level

 

Imagine for a second what this function would look like in the absence (or with a very small) barrier height. It turns out that it would end up looking very similar to the highest energy wavefunction for the n=1 band! This is pictured below:

 

PeriodicFiniteSquareWellReducedBarrier

Fig. 7: The zone boundary (\pi/a) wavefunctions for the n=1 and n=2 energy levels with a negligible barrier height

What you can see here is that at the zone boundary, the wavefunctions essentially look the same, and are essentially degenerate. This degeneracy is broken when the barriers are present.  The barriers “mess up” the wavefunction so that they no longer perfect sinusoids, changing the energies of both the zone boundary blue n=1 and the orange n=2 curves so that their energies are no longer the same. In other words, a gap has opened between the wavelength 2a n=1 and n=2 energy levels! You can sort of use your eyes to interpolate between Fig. 6 and Fig. 7 to see that the energy of the n=2 level must increase as it loses its pure sinusoidal nature and, by comparing Fig. 6 to the last image in Fig. 5, that the zone boundary wavefunction degeneracy has been lifted.

In this picture, you can also easily see that when the periodic part of the n=2 wavefunction is multiplied by the first sinusoidal function (i.e. the one with wavelength Na/2), it actually has the highest energy in the n=2 band. This can be seen by comparing the orange curves in Fig. 7 and Fig. 3. The curve in Fig. 3 has many more nodes. The lowest energy is actually obtained when the n=2 periodic function is multiplied by the sinusoidal function of wavelength 2a, i.e. at the zone boundary. This implies that in contrast to the first band, the second one disperses downward from the center of the Brillouin zone.

One more thing to note, which has been implicit in the discussion is that essentially the n=1 level has the symmetry of an s-like wavefunction whereas the n=2 level has the symmetry of a p-like wavefunction.  If one keeps going with this picture, you can essentially get d- and f-like bands as well.

I hope this post helps bring an end to the so-called “tyranny of reciprocal space”. It is not difficult to imagine the wavefunctions in real space and this framework shouldn’t be so intimidating to band theory newcomers!

I actually wonder what the limitations of this picture are — if anyone sees how to explain, for instance, the Berry phase within this picture, I’d be interested to hear it!

 

1 This of course is not strictly correct, but this helps in visualizing what is going on tremendously.

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