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As other have noted, it's discrete but with fine enough spacing to treat as continuous. However I disagree that quantum mechanics is the reason. You see the exact same thing in a classical 1-D chain o …

Your equation 1 was derived with an approximation for the Fermi-Dirac integral and was derived for 3D. That is, they used $F_{\frac{1}{2}}\left(\eta_c\right) \approx \frac{\sqrt{\pi}}{2} e^{\eta_c}$, …

You're right that this can be a tricky point if there is a gap, and I think that a number of common definitions fail in this case. It gets even trickier at zero temperature. Let me give an alternativ …

As Pieter noted, in some metals, arguably, all the carriers are holes. This can be seen by measuring a material's Hall coefficient, which tells you the effective carrier charge, and for a number of me …

The band gap of crystalline Si at 300 K is 1.12 eV. Here is an authoritative source for band structure information. Crystalline silicon is an indirect band gap semiconductor. Doping does not change th …

Right, so it turns out this is quite simple. See the figure below. (a) shows the 1st BZ resulting from the hexagonal and rectangular unit cells of the honeycomb lattice. Note that the hexagon has twi …

The heat of a solid is related to the energy contained in the solid. Hotter solids have more energy in them. Where is the energy in the solid? It's in the atoms and charge carriers moving/bouncing ar …

Both confusing diagrams show heterostructures (i.e. one device containing different materials), and the different materials can have very different properties. In the first figure: At the left is a m …

I would answer "no", but it depends what you mean. An electron going from the conduction to the valence band eliminates both a conduction band electron and a valence band hole. Both helped with conduc …

I don't think we should be so quick to say that there's no way to know without doing a calculation. In some cases, there are rules of thumb: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.98.2 …

Can the "well-known properties" be developed using free particles? (I think that the answer should be "yes".) Because the two equations are the same for free particles. For free particles: $$\epsilon\ …

Depending on what is shown in the band diagram, you can see if it's a topological insulator. Most band diagrams just show what's going on in the bulk material -- what you'd get with an infinitely larg …

Band structures do all sorts of weird things. Just look at Si if you want a conduction band that has the "wrong" concavity. That said, the concavity you describe is common. By definition, it's true fo …

You asked two separate questions: Is it possible to draw an EK diagram for the infinite square well? No. Such a diagram only makes sense if the potential is periodic, and that's not the case for an …

No, it applies whenever there are free charges. So, it applies to semiconductors too (especially doped ones) or even insulators if you can heat them up enough without melting them... EDIT: by way of e …