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There is nothing wrong with looking for plane-wave like solutions of the form $A \exp (i (\omega t - k x) )$. Given the linearity of the equations, and as @ignacio pointed out the fact that the $\exp (i k x_n)$ form a basis of solutions, you can write a more general solution as a combination of these plane waves. This solution isn't necessarily periodic ...

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One thing to note is that you are not using the primitive cell, which is obtained by taking the lattice vectors joining your A atom to the A atom diagonally down and to the right and from the original A atom diagonally up and to the right, such that the A atoms sit at the lattice points, and the B atom sits in the space between.

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The problem is that you are looking at the gap at the $\Gamma$ point ($k=0$) and not at the real gap at the Brillouin zone boundary. Here is the plot for the phonon dispersion relation of a diatomic linear chain. You can see that the real gap at the $k=\pi/a$ point is vanishing for $m_1=m_2$. This plot looks different as typical plots for a monoatomic ...

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So upon looking into this further, it looks like Cooper pair interactions are pretty weak, on the order of 10^-3 eV. As such, I'm guessing the reason that we see superconducting at low temps and not at higher temps (with a few "hot superconductors" making an exception) is because at higher temperatures there is enough energy in the system such that Cooper ...

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One way to estimate electron-phonon coupling is to take a look on the hot-electron relaxation rate, which can be more or less directly probed by pump-probe spectroscopy. The values for $\lambda$ are around 0.2-0.5. There are few articles on this topic from C. Gadermaier. Here are the links http://dx.doi.org/10.1063/1.4726164 ...

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You are probably aware that the underlying mechanism responsible for superconductivity in cuprates or Fe-based superconductors is still subject to intense debate and research. The usual BCS electron-phonon coupling (EPC) is too weak to account for superconductivity at temperatures of order $100$ K, and produces a gap of s-wave symmetry which is incompatible ...

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Phonons are the particle-like analogue of normal modes. So yes, the frequencies are the same.

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There's not enough information to say. It looks like your model has spring-like (harmonic) potentials between neighboring atoms, but phonon scattering requires anharmonic potentials. Harmonic potentials mean that the superposition principle still holds, so the phonons just pass right through each other without scattering. Since you have harmonic ...

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