Commit f1bf01fd authored by Michael Borst's avatar Michael Borst
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Update src/12_band_structures_in_higher_dimensions_solutions.md

parent 4321390e
```python tags=["initialize"]
from matplotlib import pyplot as plt
import numpy as np
from math import pi
```
# Solutions for lecture 12 exercises
## Exercise 1: 3D Fermi surfaces
### Subquestion 1
Well described: (close to) spherical.
### Subquestion 2
K is more spherical, hence 'more' free electron model. Li is less spherical, hence 'more' nearly free electron model. Take a look at Au, and see whether you can link this to what you learned in lecture 11.
### Subquestion 3
Yes. Cubic -> unit cell contains one atom -> monovalent -> half filled band -> metal.
### Subquestion 4
With Solid State knowledge: Na has 1 valence electron, Cl has 7. Therefore, a unit cell has an even number of electrons -> insulating.
Empirical: Salt is transparent, Fermi level must be inside a large bandgap -> insulating.
## Exercise 2: Tight binding in 2D
### Subquestion 1
$$ E \phi_{n,m} = \varepsilon_0-t_1 \left(\phi_{m,n-1}+\phi_{m,n+1}\right) -t_2 \left(\phi_{m-1,n}+\phi_{m+1,n}\right) $$
### Subquestion 2
$$ \psi_n(\mathbf{r}) = u_n(\mathbf{r})e^{i\mathbf{k}\cdot\mathbf{r}} \quad \leftrightarrow \quad \phi_{n,m} = \phi_0 e^{i(k_x a_x + k_y a_y)}$$
### Subquestion 3
$$ E = \varepsilon_0 -2t_1 \cos(k_x a_x) -2t_2 \cos(k_y a_y)$$
### Subquestion 4 and 5
Monovalent -> half filled bands -> rectangle rotated 45 degrees.
Much less than 1 electron per unit cell -> almost empty bands -> circular.
```python
def dispersion2D(N=100, kmax=pi, e0=2):
# Define matrices with wavevector values
kx = np.tile(np.linspace(-kmax, kmax, N),(N,1))
ky = np.transpose(kx)
# Plot dispersion
plt.figure(figsize=(6,5))
plt.contourf(kx, ky, e0-np.cos(kx)-np.cos(ky))
# Making things look ok
cbar = plt.colorbar(ticks=[])
cbar.set_label('$E$', fontsize=20, rotation=0, labelpad=15)
plt.xlabel('$k_x$', fontsize=20)
plt.ylabel('$k_y$', fontsize=20)
plt.xticks((-pi, 0 , pi),('$-\pi/a$','$0$','$\pi/a$'), fontsize=17)
plt.yticks((-pi, 0 , pi),('$-\pi/a$','$0$','$\pi/a$'), fontsize=17)
dispersion2D()
```
## Exercise 3: Nearly-free electron model in 2D
### Subquestion 1
Construct the Hamiltonian with basis vectors $(\pi/a,0)$ and $(-\pi/a,0)$, eigenvalues are
$$ E=\frac{\hbar^2}{2m} \left(\frac{\pi}{a}\right)^2 \pm \left|V_{10}\right|^2. $$
### Subquestion 2
Four in total: $(\pm\pi/a,\pm\pi/a)$.
### Subquestion 3
Define a basis, e.g.
\begin{align}
\left|0\right\rangle &= (\pi/a,\pi/a) \\
\left|1\right\rangle &= (\pi/a,-\pi/a) \\
\left|2\right\rangle &= (-\pi/a,-\pi/a) \\
\left|3\right\rangle &= (-\pi/a,\pi/a)
\end{align}
The Hamiltonian becomes
$$
\hat{H}=
\begin{pmatrix}
\varepsilon_0 & V_{10} & V_{11} & V_{10} \\
V_{10} & \varepsilon_0 & V_{10} & V_{11} \\
V_{11} & V_{10} & \varepsilon_0 & V_{10} \\
V_{10} & V_{11} & V_{10} & \varepsilon_0 \\
\end{pmatrix}
$$
### Subquestion 4
$$ E = \varepsilon_0 + V_{11} \quad \text{and}\quad E = \varepsilon_0 - V_{11} \pm \left|V_{10}\right|^2 $$
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