The Geometry of Band Structure

This article relates to Web Chapter 1 on The Band Theory of Solids

I received a delightfully-worded request for clarification from a correspondent in India the other day. I say “delightfully-worded” because the correspondent started out by complimenting me :-).

The enquiry concerned a subtle detail of the band theory of solids that requires an understanding of the way the band gap affects electronic states in two or three dimensions.

This extension beyond one dimension is particularly difficult to visualise.

Anyway: here is the letter: my response is below.

Dear Michael,

I have read the web chapter on the band theory of solids in physicsofmatter. I really enjoyed reading for its clarity of expression.

It was most elegantly presented than elsewhere on web or books. Thank you for creating such a wonderful document.

But, I could not follow the following paragraph:

Figure W1.15 shows how the energy varies as one proceeds from the origin along lines joining the origin to the point (0, ¹/a) and to the point ( ¹/a, ¹/a). Notice that the magnitude of the parameter Vo in Equation 1.80 now becomes very important.

If Vo is large compared with the maximum energy of a quantum state in the first Brillioun zone, then all the quantum states in the second Brillioun Zone will have energies higher than any quantum state in the first Brillioun Zone. In this situation there is what is known as a band gap between states in the two zones. · 

However, if Vo is small compared with the maximum energy of a quantum state in the first Brillioun zone, then some of the quantum states in the second Brillioun Zone will have energies lower than some quantum states in the first Brillioun Zone. In this situation there is said to be band overlap between states in the two zones. 

The point I did not understand is  precisely this “second Brillioun Zone will have energies lower than some quantum states in the first Brillioun Zone. In this situation there is said to be band overlap between states in the two zones.”  I do not  see any band overlap in the figure or see any quantum states in second zone whose energy is lower than the first zone. I know, I am missing something but can not figure out what it is? Can you please help me in understanding this point?

The letter refers to Figure W1.15 which looks like this:

Figure W1.15 Illustration of the results of two-dimensional, nearly-free electron calculation for the case of two electrons per lattice site. (a) (b) and (c) correspond to the case when V0 is large and (d) (e) and (f) to the case when V0 is small. (a) and (d) show the structure of E(k) along k,0,0 and k,k,0 directions (b) and (e) show the resulting density of states. Notice that in (b) the density of states at the Fermi energy is zero which is indicative of insulator. (c) and (f) show three dimensional views of the band structure in the two bands
Figure W1.15 Illustration of the results of two-dimensional, nearly-free electron calculation for the case of two electrons per lattice site. (a) (b) and (c) correspond to the case when V0 is large and (d) (e) and (f) to the case when V0 is small. (a) and (d) show the structure of E(k) along k,0,0 and k,k,0 directions (b) and (e) show the resulting density of states. Notice that in (b) the density of states at the Fermi energy is zero which is indicative of insulator. (c) and (f) show three dimensional views of the band structure in the two bands

The key point can be seen if we enlarge part (f) of the above figure:

Part (f) of the figure above. The key point is that if the splitting between the upper and lower band is small, then the lowest point in the upper band (yellow square) is higher in energy the highest point in the lower band (yellow circles).
Part (f) of the figure above with key points highlighted.

The key point is that if the splitting between the upper and lower band is small, then the lowest point in the upper band (yellow square) is lower in energy than the highest point in the lower band (yellow circle).

The perspective of the drawing means that I can’t clearly show all the symmetrically equivalent points in this diagram. So I have shown only one of the four equivalent points in the upper band (square), and three of the four equivalent points in the lower band (circles)

Because of this, a material with such a band structure would have electronic states (occupied or unoccupied) at every energy, so there would be no energy gap. This is illustrated in part (e) of Figure W1.15.

This contrasts with the case shown in part (c)  of Figure W1.15

Part (c)
Part (c)

Now we can see that if the splitting between the upper and lower band is large, then the lowest points in the upper band (yellow square) are higher in energy than even the highest point in the lower band (yellow circles).

Because of this, in a material with such a band structure would there would be an energy gap where there were no states that could be occupied by electrons. This is illustrated in part (b) of Figure W1.15.

A reminder of why this matters.

The reason that this kind of analysis matters is that it allows us to explain astonishing qualitative differences between materials in terms of a single quantitative parameter – in this case Vo.

So for example argon and potassium are spectacularly different materials. We know from §6.2 and §7.3, that argon is a gas at room temperature and forms a ‘molecular solid’ at low temperatures.

However, when we describe potassium, we would naturally discount the theory for argon as irrelevant, and use the theory for a free electron gas described in §6.5.

But argon and potassium atoms differ by only a single electron in their outer electronic shells. Can we find a theory that explains why the addition of a single electron per atom makes potassium a metal? This is what the band theory of solids does.

The band theory of solids is essentially a framework that allows us to discuss the nature of electronic states in all kinds of solids, and to explain from first principles why one type of atom forms solids that are insulating, and other similar atoms form solids that are metals.

This is an astounding achievement.

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