 ### A bit of reformulation

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 ... ... @@ -209,12 +209,12 @@ The spacing of the distorted chain alternates between two different distances an Due to the alternating hopping energies, we must treat two consecutive atoms as two different orbitals ($\left| n,1 \right>$ and $\left| n,2 \right>$ in the figure) from the same unit cell. The corresponding LCAO of this chain is given by $$\left|\Psi \right> = \sum_n \left( \phi_n \left| n,1 \right> + \psi_n \left| n,2 \right>\right).$$ As usual, we assume that all these atomic orbitals are orthogonal to each other. 1. Indicate the length of the unit cell $a$ in the figure. 2. Give the expressions of the tight binding Hamiltonian. 2. Using the Schrödinger equation, write the equations of motion of the electrons. ??? hint To this end, find expressions for $E \left< n,1 \vert \Psi \right> = \left< n,1 \right| H \left|\Psi \right>$ and $E \left< n,2 \vert \Psi \right> = \left< n,2 \right| H \left|\Psi \right>$. 3. Using the trial solutions $\phi_n = \phi_0 e^{ikna}$ and $\psi_n = \psi_0 e^{ikna}$, show that the Schödinger equation can be rewritten in matrix form: $$\begin{pmatrix} \epsilon & t_1 + t_2 e^{-i k a} \\ t_1 + t_2 e^{i k a} & \epsilon \end{pmatrix} \begin{pmatrix} \phi_0 \\ \psi_0 \end{pmatrix} = E \begin{pmatrix} \phi_0 \\ \psi_0 \end{pmatrix}.$$ 3. Using the trial solutions $\phi_n = \phi_0 e^{ikna}$ and $\psi_n = \psi_0 e^{ikna}$, show that the Schödinger equation can be written in matrix form: $$\begin{pmatrix} \epsilon & t_1 + t_2 e^{-i k a} \\ t_1 + t_2 e^{i k a} & \epsilon \end{pmatrix} \begin{pmatrix} \phi_0 \\ \psi_0 \end{pmatrix} = E \begin{pmatrix} \phi_0 \\ \psi_0 \end{pmatrix}.$$ 4. Derive the dispersion relation of this Hamiltonian. Does it look like the figure of the band structure shown on the [Wikipedia page](https://en.wikipedia.org/wiki/Peierls_transition#/media/File:Peierls_instability_after.jpg)? Does it reduce to the 1D, equally spaced atomic chain if $t_1 = t_2$? 5. Find an expression of the group velocity $v(k)$ and effective mass $m^*(k)$ of both bands. 6. Derive an expression for the density of states $g(E)$ of the entire band structure and make a plot of it. Does your result makes sense when considering the band structure? ... ...
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