# Bands in semiconductors: $E$ vs. $k$ diagram

I always thought that in a semiconductor there was one gap, one conduction band and one valence band. However, reading a book I came across this picture

And now I'm very confused. Apparently there aris an infinite number of gaps (i.e. forbidden energy bands). Is this so? And what about the conduction and valence bands? Which of all the bands in the graphic corresponds to them?

Also, do all the electrons in the conduction or valence band have the same energy? Or can they differ? Because from this graphic, it appears that electrons with different $k$ can belong to the same band but have different amounts of energy $E$. Is this correct or not?

## 1 Answer

There is one relevant gap in semiconductor physics: the gap between the highest fully occupied energy band (valence band) and the lowest unoccupied band (conduction band). But there are also gaps between other energy bands. The thing is just that they are not very relevant for the semiconducting effect. Between full bands they have no effect because there is no transition possible between full bands and between empty bands well ... they are empty anyways, aren't they? As to which band in the graphic corresponds to which band, this depends on the semiconductor we are talking about. Note that the energy bands (or surfaces/volumes in 3D k-space) of real materials often look a lot more complicated. Electrons can belong to the same band and have different energies, that is correct. Each band represents an intervall of possible energies that the electrons in this band can take on. But as there is an energy gap between conduction and valence band, an electron in the conduction band has at least the amount of that energy gap more energy than the electron in the valence band.

• In the case of silicon, I understand that its energy levels 1 and 2 are fully occupied, so the energy level 3 would be the valence one, right? Does this mean that the first two allowed bands are full and the third one (corresponding to $2\pi /a < |k|<3\pi /a$) would be the valence band? If this is right, then I believe the next band will be the conduction band. And what about the next bands? Can electrons go up there? Do they exist or what? Jun 15, 2016 at 22:22
• The real case is unfortunately not as simple as to just have numbered energy levels. If you have access to an Ibach Lüth Solid State Physics you can see the scheme for Si in chapter 12. But the principle does not change. The highest fully occupied one will be the valence band. The empty bands above the conduction band "exist" but a lot of energy would be needed to arrive there and an electron would probably go down again into a lower band pretty soon. Jun 15, 2016 at 22:34
• So would that mean that what we call the "conduction band" is actually the first band that comes after the valence one? Jun 15, 2016 at 22:36
• yes, simply because if electrons are energised into a higher band, it is most probable for them to land there AND they can already move in that band (as opposed to the full bands below) Jun 15, 2016 at 22:38
• I might suggest that the OP is also confused because the plot they reproduced is not in a reduced-zone. Once everything is brought back into the first zone it becomes easier to see that it looks more like the typical "band" structure pictures. Although, for devices, band structure is put against position in the device, or just against no real axis to draw in the dopants/acceptors/whatnot. Actual plots of E vs k along different directions are usually glossed over in intro semiconductor courses (and definitely in device courses). Jun 15, 2016 at 23:00