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http://onlinelibrary.wiley.com/doi/10.1002/pssr.201307003/full

Dimensionally you are correct. But in this case, unfortunately, you have to perform the detailed calculus steps in order to get that factor. First let us determine the expression for ##n##. In ##\bf k##-space you need to count the total number of occupied states. This can be computed as seen in...

Yes, the ##\Delta K## you've shown in the updated figure is correct. Yes, if you do it the numerical calculations correctly you should get the exact solution. Now, comparing the exact solution to the perturbative one is a little tricky. In the Kronig-Penney model, which is discussed in the...

No, that is not the correct ##\Delta K## (##= K_1 - K_2##). That ##\Delta K## corresponds to range of energies ##\Delta E = \hbar^2 K_1^2/2m - \hbar^2 K_2^2/2m## that are allowed in the dispersion relation. You want to determine the ##\Delta E## for which the states are forbidden. That region...

This approach more or less seems like the one adopted by the link I pasted in my first post: http://faraday.ee.emu.edu.tr/eeng245/KronigPenney.pdf I don't think that's making use of perturbation theory; this seems like an exact calculation. If you look at the nearly-free electron model then...

As DrDu correctly pointed out, you do not need SOC to get topological order. However, SOC is indeed a necessary ingredient in HgTe and many other materials which have so far been experimentally confirmed as "band" topological insulators. By "band" topological insulators I mean materials in which...

Yes, you can associate ##k## as the wave vectors of phonons. Lattice vibrations give an intuitive feel for what ##k## represents in that specific context. However, ##k## can be defined much more generally. Similar to how you have quantum numbers ##(n,l,m,s)## (principal, azimuthal, magnetic...

This lightscribe method appears to be one of the large-area graphene production methods. Although, in the video he does not explain how he made the super capacitor. To me it seems like he simply cut out a rectangular section of graphene from the disc and made a capacitor (without any kind of...

The (lowercase) ##k## is the so-called crystal momentum. Say your system contains ##N## such copies of the potential profile that is being repeated after every ##a## distance. Then using the Born-von Karman boundary conditions you will get ##k = m\frac{2\pi}{Na}## where ##m## goes from ##0##...

No, the scotch tape technique is only used for research. This technique is extremely painful and has a terrible yield. Not to mention that the graphene flake sizes you would get would not be longer then a couple of tens of microns. When the properties of graphene were reported for the first time...

You seem to have the correct picture for the first two diagrams in terms of the flow of carriers and the direction band bending. In the third diagram, however, it doesn't make sense to talk about flow of holes. Since in the metal there is no electronic excitation gap it does not make sense to...

Equation (8.63) from Ashcroft and Mermin will be helpful. I have listed it below for convenience: ##g_n(\mathcal{E})=\int_{S_n(\mathcal{E})} \frac{dS}{4\pi^3}\frac{1}{|\nabla\mathcal{E}_n(k_x,k_y,k_z)|}## I know you want to know how to compute density of states numerically. I was referring to...

What property is it that you're trying to study? One example is the band gap at the ##\textbf{K}## and ##\textbf{K}^\prime## points. I can think of one reference where the gap at these points this discussed as a function of buckling of the honeycomb lattice. That reference is...

Is this something worth looking into: http://prola.aps.org/abstract/PR/v52/i4/p365_1?

In a crystal you label the Bloch states using ##\textbf{k}##. Therefore ##\textbf{k}## is a quantum number (or numbers if you count the three components) (pg. 141 of Ashcroft and Mermin). So yes, at the intersection point the quantum numbers are in fact the same. Also, can you please provide...