@@ -235,16 +235,16 @@ Recall the eigenfrequencies of a diatomic vibrating chain in the lecture notes w
#### Exercise 2: atomic chain with 3 different spring constants
Suppose we have a vibrating 1D atomic chain with 3 different spring constants alternating like $\kappa_ 1$, $\kappa_2$, $\kappa_3$, $\kappa_1$, etc. All the the atoms in the chain have an equal mass $m$.
1. Make a sketch of this chain and indicate the length of the unit cell $a$ in this sketch.
2. Derive the equations of motion for this chain.
3. By filling in the trial solutions for the equations of motion (which should be similar to the ones in eqs. (10.3) and (10.4) of the book), show that the eigenvalue problem is given by $$ \omega^2 \mathbf{x} = \frac{1}{m} \begin{pmatrix} \kappa_1 + \kappa_ 3 & -\kappa_ 1 & -\kappa_ 3 e^{i k a} \\ -\kappa_ 1 & \kappa_1+\kappa_2 & -\kappa_ 2 \\ -\kappa_ 3 e^{-i k a} & -\kappa_2 & \kappa_2 + \kappa_ 3 \end{pmatrix} \mathbf{x}$$
4. In general, the eigenvalue problem above cannot be solved analytically, and can only be solved in specific cases. Find the eigenvalues $\omega^2$ when $k a = \pi$ and $\kappa_ 1 = \kappa_ 2 = q$.
1. Make a sketch of this chain and indicate the length of the unit cell $a$ in this sketch.
2. Derive the equations of motion for this chain.
3. By filling in the trial solutions for the equations of motion (which should be similar to the ones in eqs. (10.3) and (10.4) of the book), show that the eigenvalue problem is given by $$ \omega^2 \mathbf{x} = \frac{1}{m} \begin{pmatrix} \kappa_1 + \kappa_ 3 & -\kappa_ 1 & -\kappa_ 3 e^{i k a} \\ -\kappa_ 1 & \kappa_1+\kappa_2 & -\kappa_ 2 \\ -\kappa_ 3 e^{-i k a} & -\kappa_2 & \kappa_2 + \kappa_ 3 \end{pmatrix} \mathbf{x}$$
4. In general, the eigenvalue problem above cannot be solved analytically, and can only be solved in specific cases. Find the eigenvalues $\omega^2$ when $k a = \pi$ and $\kappa_ 1 = \kappa_ 2 = q$.
??? hint
??? hint
To solve the eigenvalue problem quickly, make use of the fact that the Hamiltonian in that case commutes with the matrix $$ X = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}. $$ What can be said about eigenvectors of two matrices that commute?
To solve the eigenvalue problem quickly, make use of the fact that the Hamiltonian in that case commutes with the matrix $$ X = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}. $$ What can be said about eigenvectors of two matrices that commute?
5. What will happen to the periodicity of the band structure if $\kappa_ 1 = \kappa_ 2 = \kappa_3$?
5. What will happen to the periodicity of the band structure if $\kappa_ 1 = \kappa_ 2 = \kappa_3$?
#### Exercise 3: the Peierls transition
In the previous lecture, we have derived the electronic band structure of an 1D, equally spaced atomic chain. Such chains, however, are in fact not stable and the equal spacing will be distorted. This is also known as the [Peierls transition](https://en.wikipedia.org/wiki/Peierls_transition).
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@@ -255,14 +255,14 @@ 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 different orbitals ($\left| \phi_n \right>$ and $\left| \psi_n \right>$ in the figure) from the same unit cell. This also means that we expect to find two eigenvalues for the Schrödinger equation and we also need two different LCAO's to form a basis for the Hilbert space. These LCAO's are $$\left| \Phi\right>= \sum_n a_n \left| \phi_n \right>,$$ $$\left| \Psi\right>= \sum_n b_n \left| \psi_n \right>,$$ and the coefficients $a_n$ and $b_n$ are both given by $$a_n = b_n=\frac{e^{-ikna}}{\sqrt{N}},$$ where $N$ is the total number of unit cells in the (periodic) chain.
1. Indicate the length of the unit cell in the figure
2. What are the values of the matrix elements $\left<\phi_n\right|H\left|\phi_{n}\right>$, $\left<\psi_n\right|H\left|\psi_{n}\right>$, $\left<\psi_n\right|H\left|\phi_{n}\right>$, and $\left<\psi_n\right|H\left|\phi_{n+1}\right>$?
3. Using the $\{\left| \Phi\right>, \left| \Psi\right> \}$-basis, show that the Hamiltonian is given by $$H = \begin{pmatrix} \epsilon & t_1 + t_2 e^{i k a} \\ t_1 + t_2 e^{-i k a} & \epsilon \end{pmatrix}.$$
1. Indicate the length of the unit cell in the figure
2. What are the values of the matrix elements $\left< \phi_n \right| H \left| \phi_{n} \right>$, $\left< \psi_n \right| H \left| \psi_{n} \right>$, $\left< \psi_n \right| H \left| \phi_{n} \right>$, and $\left< \psi_n \right| H \left| \phi_{n+1} \right>$?
3. Using the $\{ \left| \Phi\right>, \left| \Psi\right> \}$-basis, show that the Hamiltonian is given by $$H = \begin{pmatrix} \epsilon & t_1 + t_2 e^{i k a} \\ t_1 + t_2 e^{-i k a} & \epsilon \end{pmatrix}.$$
??? hint
??? hint
Note that the coefficients of the trial LCAO's and the periodic boundary condition require that $\left<\Phi\right|H\left|\Phi\right> = N a_n \left<\phi_n\right|H\left|\Phi\right>$ for any $n$. Similar can be done for $\left<\Phi\right|H\left|\Psi\right>$ and $\left<\Psi\right|H\left|\Psi\right>$.
Note that the coefficients of the trial LCAO's and the periodic boundary condition require that $\left< \Phi \right| H \left| \Phi \right> = N a_n \left< \phi_n \right| H \left| \Phi \right>$ for any $n$. Similar can be done for $\left< \Phi \right| H \left| \Psi \right>$ and $\left< \Psi \right| H \left| \Psi \right>$.
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.
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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.
