@@ -222,3 +222,45 @@ $\rightarrow \rho_{\rm R}(k)=\frac{L}{2\pi}$, which is lower than for the case o
* Computing dispersions explains all the problems we listed before (need for cutoff, lack of scattering with every single atom, existence of insulators).
* Electrons and phonons have a complicated nonlinear relation between momentum and velocity (**group velocity**), effective mass, and density of states
* In a system with more than one degree of freedom per unit cell we need to consider independent amplitudes for each degree of freedom, and get multiple bands.
## Exercises
#### Exercise 1: analyzing the diatomic vibrating chain
Recall the eigenfrequencies of a diatomic vibrating chain in the lecture notes with 2 different masses (can be found below [here](#more-degrees-of-freedom-per-unit-cell)).
1. Find the magnitude of the group velocity near $k=0$ for the _acoustic_ branch.
2. Show that the group velocity at $k=0$ for the _optical_ branch is zero.
3. Derive an expression for the density of states $g(\omega)$ in the _optical_ branch.
4. Make a plot of your expression of $g(\omega)$ found in 3. Does the plot look like the bar diagram of the density of states of the optical branch in the lecture notes?
#### Exercise 2: the 1D tight binding model with 2 orbitals
In chapter 11 of the book, the equation of motion for 1D tight binding chain for 1 atom per unit cell has been derived. In this exercise, we are going to do this for a chain with 2 different atoms per unit cell as depicted below.
We have two types of orbitals $\left| \phi_n \right>$ and $\left| \psi_n \right>$ belonging to atom A and B respectively. The onsite energies are $\epsilon_A$ and $\epsilon_B$ for atom A and B respectively. Between the alternating A and B atoms, there is a distance $a/2$ and the hopping between them is given by $\left<\psi_{n-1}\right|H\left|\phi_n\right> = \left<\phi_n\right|H\left|\psi_{n}\right> = \left<\psi_n\right|H\left|\phi_{n+1}\right> = t$, where $t$ will be assumed to be real. Futhermore, it is assumed that the all the orbitals are orthonormal to each other.
Because there are two different atoms in the chain, we expect to find two eigenvalues for the Schödinger equation. We also need two different LCAO's to form a basis. 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 A _or_ B atoms in the chain.
1. Using the $\{\left| \Phi\right>, \left| \Psi\right> \}$-basis and periodic boundary conditions for the chain, show that the Hamiltonian is given by $$H = \begin{pmatrix} \epsilon_A & t \left[ 1 + e^{i k a}\right] \\ t \left[ 1 + e^{-i k a}\right] & \epsilon_B \end{pmatrix}$$ **Hint:** note that the coefficients of the trial LCAO's 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 $\left<\Psi\right|H\left|\Psi\right>$.
2. Find the eigenvalues of this Hamiltonian. Make a plot of the band structure using your expression of these eigenvalues.
3. What will happen to the periodicity of the band structure if $\epsilon_A=\epsilon_B$?
#### Exercise 3: 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 eq. (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:** 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 $\kappa_ 1 = \kappa_ 2 = \kappa_3$?
#### Exercise 4: 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).
The spacing of the distorted chain alternates between two different distances $a_1$ and $a_ 2$ and this also causes the hopping energy to alternate between $t_1$ and $t_2$ (check also [this figure](https://en.wikipedia.org/wiki/Peierls_transition#/media/File:Peierl%27sDistortion1-D.png) on Wikipedia). We further set the onsite energies of the atoms to $\epsilon$.
1. Make a sketch of the chain and indicate the unit cell length $a$ in this sketch.
2. Using a similar procedure as in exercise 2, show that the Hamiltonian of this system is given by $$H = \begin{pmatrix} \epsilon & t_1 + t_2 e^{i k a} \\ t_1 + t_2 e^{-i k a} & \epsilon \end{pmatrix}.$$
3. Derive the dispersion relation of this Hamiltonian. Does it look like [this figure](https://en.wikipedia.org/wiki/Peierls_transition#/media/File:Peierls_instability_after.jpg) shown on Wikipedia? Does it reduce to the 1D, equally spaced atomic chain if $t_1 = t_2$.
4. Find an expression of the group velocity $v(k)$ and effective mass $m^*(k)$ of both bands.
5. Derive an expression for the density of states $g(E)$ of the entire band structure.
