12_band_structures_in_higher_dimensions_solutions.md

search:
  exclude: true

Solutions for lecture 12 exercises

from matplotlib import pyplot as plt 
import numpy as np
from math import pi

Exercise 1: 3D Fermi surfaces

Question 1.

Well described: (close to) spherical.

Question 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.

Question 3.

Yes. Cubic -> unit cell contains one atom -> monovalent -> half filled band -> metal.

Question 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

Question 1.

E \phi_{n,m} = \varepsilon_0\phi_{n,m}-t_1 \left(\phi_{n-1,m}+\phi_{n+1,m}\right) -t_2 \left(\phi_{n,m-1}+\phi_{n,m+1}\right)

Question 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 n a_x + k_y m a_y)}

Question 3.

E = \varepsilon_0 -2t_1 \cos(k_x a_x) -2t_2 \cos(k_y a_y)

Question 4 and 5.

Monovalent -> half filled bands -> rectangle rotated 45 degrees.

Much less than 1 electron per unit cell -> almost empty bands -> elliptical.

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

Question 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|.

Question 2.

Four in total: (\pm\pi/a,\pm\pi/a).