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I'm having a bit of trouble understanding the semiconductor band gap diagram on Wikipedia: enter image description here (from Band gap article).

Why is the size of the band gap increasing with the Density of States (DOS) in the semiconductor material? I would expect the opposite given the intuiton that more occupiable energy states would mean a higher probability of an electron moving from a valence band to a conduction band?

Should the x-axis here maybe be something like "# electrons in the conducting band" to show the increasing amount of energy needed to promote electrons to the conduction band as a function of the electrons already promoted to the conduction band? And why the ovoid geometry, which I've seen elsewhere?

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You shouldn't see the density of states as a variable (the graph can be a bit misleading when plotted as it is). It would probably be better to see the graph with the axis interchanged.

There, you have bands with certain width and a density of states. The band gap is defined as the energy between the borders of the valence and conduction bands, where the DOS is vanishing. This is the minimum energy needed to promote an electron from the low energy band to the higher one. With higher energies, you can promote electrons deeper into the valence band and the effect will be stronger since the DOS is higher (aka more electrons available).

Thus, the band gap does not increase and your intuition is partially correct.

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  • $\begingroup$ Everything you've just said makes perfect sense to me. However, why are conduction and valance bands always drawn as ovoids in this manner? Even if we interchange the axes to properly represent the increasing amount of energy we need to provide to jump the gap as the distance between the valence and conduction bands increases, there's still no apparent reason to use these ovoids? $\endgroup$ – user36056 Dec 19 '13 at 10:22
  • $\begingroup$ Should the $x$-axis here maybe be function of the increasing amount of energy needed to promote electrons to the conduction band as a function of the electrons already promoted to the conduction band? $\endgroup$ – user36056 Dec 19 '13 at 10:30
  • $\begingroup$ Simplicity. It's just a picture to convey the main mechanism and ideas. Real material are messy in their DOS and in simple theoretical models the valence band looks like that and the conduction band is expected to have a $(E-E_g)^{\frac{1}{2}}$ behavior. And no, the x-axis is just energy to describe the DOS of the system. Then you realize, "ha! There is a gap with a certain size." $\endgroup$ – Ignacio Vergara Kausel Dec 19 '13 at 10:35
  • $\begingroup$ What do you mean "the x-axis is just energy to describe the DOS of the system"? Shouldn't the density of states be independent of system energy in the sense that we're not talking about occupied states? $\endgroup$ – user36056 Dec 19 '13 at 10:41
  • $\begingroup$ Yeah, so this is cycling back to my point of confusion... my intuition says flip the orientation of the arrow for the $x$-axis, since increasing the number of conduction band states should increase the probability of an electron jumping to some conducting band (if we could write down something like a partition function). Also, it makes sense to me when you say that there is going to be some messy landscape of gaps and that this is a simplified model. But clearly there is some attempt to convey something about DOS here, right? $\endgroup$ – user36056 Dec 19 '13 at 10:53
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In the graph,

  • Electron energy is the independent variable

  • Density of states is the dependent variable

i.e., "density of states is a function of electron energy".

For 99% of graphs that you've ever seen in your life, the independent variables is plotted on the x-axis and the dependent variable on the y-axis. But for this graph, it's the opposite!

Here, I shall now rotate the graph so that the independent variable is on the x-axis, like it is in most graphs :-D

Rotated plot

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