Update src/12_band_structures_in_higher_dimensions_solutions.md

src/12_band_structures_in_higher_dimensions_solutions.md

+```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
+
+$$ E = \varepsilon_0 + V_{11} \quad \text{and}\quad E = \varepsilon_0 - V_{11} \pm \left|V_{10}\right|^2 $$
+
+