# Understanding electronic band structure diagrams

Currently I'm trying to understand electronic band structures such as depicted below:

band structure http://ej.iop.org/images/1367-2630/14/3/033045/Full/nj413738f1_online.jpg

And following questions were arisen.

1. Why are there multiple lines in valence side and conduction side? Where are the bands and gaps between them starting from the lowest energy (inner electrons) to higher energies (up to valence band and conduction band)? How can I distinguish between them in the pictures like presented above? I just want to see a connection of that picture with the following: band structure http://www4.nau.edu/meteorite/meteorite/Images/ConductionBand.jpg
2. Why do these different lines intersect each other at some points? (Don't they?) What does it mean?
3. Why do we choose path connecting the points of high symmetry in 1-st Brillouine zone? What's wrong with random directions? Does this path cover all possible energy values of electron in crystal? If so, then how come is that?

Thanks in advance!

## 1 Answer

Your second figure is a simplification of the first one, usually in the $\Gamma$ point, but it could be any other as well.

Regarding your questions: There are multiple lines in valence and conduction band because there are several allowed bands or energy eigen states. Technically there is even an infinite number of allowed bands, but usually you would only plot the lowest ones, which are actually populated.

From this diagram, it seems that the lowest bandgap is at the L point.

These lines can intersect if there's multiple bands, which happen to have the same energy in a certain point.

The fixed paths in the band diagram (e.g. $\Gamma$ to M or $\Gamma$ to L are just simplifications that let you estimate the material behavior. You could move along any path, but since your carriers usually populate one of the valleys, you're only interested in a small region around a local conduction band minimum or valence band maximum.

• Thanx for the reply. But even more questions appeared. 1. So different lines are just solutions of a quantum-mechanical equation? Then how can I recognize which band (i.e., eigenstate) has the lowest quantum number? How can I order the lines looking just at diagram from the first eigen state to the higher ones? 2. Do the different lines somehow connected to s-, p-, d-, f- hybridizations and both heavy and light holes? How? 3. By "lowest bandgap" you meant "lowest band"? Cause "bandgap" has its strict definition.. If the second ("band") is the case, then how did you realize that? – Capo Pavel Mestre Apr 29 '15 at 11:50
• The different lines are so to say different solutions of Schrödinger's equation (in fact slightly different equations due to different atom spacing and therefore resulting potential for the different directions). Lowest bandgap to me means the bandgap with the lowest energy difference between VB maximum and CB minimum (which could be indirect and therefore not coincide on the same reciprocal lattice point). With the convention to put the origin of the energy axis to the VB maximum, the lowest bandgap is quite obviously $\Gamma-L$ in your first diagram. – engineer Apr 30 '15 at 6:48
• Thanks. So, if I get right, each line represents the whole band, right? Then, what is the level of degeneracy in terms of known quantum numbers s,l,j,m? – Capo Pavel Mestre May 1 '15 at 6:01
• Each line represents a band at special points or along a path in reciprocal space. The degeneracy of these bands is not included here. You would need this as extra information, as far as I know. – engineer May 1 '15 at 7:26