Commit f1bf01fd by Michael Borst

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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