@@ -232,35 +232,35 @@ Recall the eigenfrequencies of a diatomic vibrating chain in the lecture notes w
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@@ -232,35 +232,35 @@ Recall the eigenfrequencies of a diatomic vibrating chain in the lecture notes w
3. Derive an expression for the density of states $g(\omega)$ in the _optical_ branch.
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?
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
#### Exercise 2: atomic chain with 3 different spring constants
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 $m$.
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.
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.
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 $$ \omega^2 \mathbf{x} = \frac{1}{m} \begin{pmatrix} \kappa_1 + \kappa_ 3 & -\kappa_ 1 & -\kappa_ 3 e^{i k a} \\ -\kappa_ 1 & \kappa_1+\kappa_2 & -\kappa_ 2 \\ -\kappa_ 3 e^{-i k a} & -\kappa_2 & \kappa_2 + \kappa_ 3 \end{pmatrix} \mathbf{x}$$
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:** 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?
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 $\kappa_ 1 = \kappa_ 2 = \kappa_3$?
5. What will happen to the periodicity of the band structure if $\kappa_ 1 = \kappa_ 2 = \kappa_3$?
#### Exercise 4: the Peierls transition
#### 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).
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 $\epsilon$.
The spacing of the distorted chain alternates between two different distances and this also causes the hopping energy to alternate between $t_1$ and $t_2$. We further set the onsite energies of the atoms to $\epsilon$. The situation is depicted in the figure below.
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 $$H = \begin{pmatrix} \epsilon & t_1 + t_2 e^{i k a} \\ t_1 + t_2 e^{-i k a} & \epsilon \end{pmatrix}.$$
??? 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 $\left<\Psi\right|H\left|\Psi\right>$.
1. Make a sketch of the chain and indicate the unit cell length $a$ in this sketch.
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 $t_1 = t_2$.
2. Using a similar procedure as in exercise 2, show that the Hamiltonian of this system is given by $$H = \begin{pmatrix} \epsilon & t_1 + t_2 e^{i k a} \\ t_1 + t_2 e^{-i k a} & \epsilon \end{pmatrix}.$$
5. Find an expression of the group velocity $v(k)$ and effective mass $m^*(k)$ of both bands.
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 $t_1 = t_2$.
6. Derive an expression for the density of states $g(E)$ of the entire band structure.
4. Find an expression of the group velocity $v(k)$ and effective mass $m^*(k)$ of both bands.
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5. Derive an expression for the density of states $g(E)$ of the entire band structure.
